QUANTUM INFORMATION PROCESSING WITH AN ASYMMETRIC ERROR CHANNEL

CROSS REFERENCE TO RELATED APPLICATIONS

[0001]This application claims the benefit under 35 U.S.C. § 119(e) of U.S. Provisional Patent Application No. 62/692,243, filed Jun. 29, 2018, titled “FAULT TOLERANT MEASUREMENTS AND GATES FOR QUANTUM INFORMATION PROCESSING,” which is hereby incorporated by reference in its entirety.

FIELD

[0002]The technology described herein relates generally to quantum information systems. Specifically, the present application is directed to systems and methods for performing quantum information processing (QIP) using at least one qubit with an asymmetric error channel.

STATEMENT REGARDING FEDERALLY-SPONSORED RESEARCH AND DEVELOPMENT

[0003]This invention was made with government support under 1609326 awarded by National Science Foundation, support under FA9550-15-0029 awarded by United States Air Force Office of Scientific Research, support under N00014-16-2270 awarded by United States Office of Naval Research and support under W911NF-14-1-0011 and W911NF-16-1-0349 awarded by United States Army Research Office. The government has certain rights in the invention.

BACKGROUND

[0004]QIP uses quantum mechanical phenomena, such as energy quantization, superposition, and entanglement, to encode and process information in a way not utilized by conventional information processing. For example, it is known that certain computational problems may be solved more efficiently using quantum computation rather than conventional classical computation. However, to become a viable computational option, quantum computation requires the ability to precisely control a large number of quantum bits, known as “qubits,” and the interactions between these qubits. In particular, qubits should have long coherence times, be able to be individually manipulated, be able to interact with one or more other qubits to implement multi-qubit gates, be able to be initialized and measured efficiently, and be scalable to large numbers of qubits.

[0005]A qubit may be formed from any physical quantum mechanical system with at least two orthogonal states. The two states of the system used to encode information are referred to as the “computational basis.” For example, photon polarization, electron spin, and nuclear spin are two-level systems that may encode information and may therefore be used as a qubit for QIP. Different physical implementations of qubits have different advantages and disadvantages. For example, photon polarization benefits from long coherence times and simple single qubit manipulation, but suffers from the inability to create simple multi-qubit gates.

[0006]The above examples of qubits are physical two-level systems. However, quantum information may also be stored in logical qubits, which are formed from multiple physical two-level systems or quantum systems with more than two states. For example, states of a quantum mechanical oscillator, of which there are an infinite number of energy eigenstates, may also be used to form the computational basis for QIP. For example, coherent states of a quantum mechanical oscillator that are sufficiently displaced from one another in phase space are quasi-orthogonal states and may be used as a computational basis. Additionally, states that are superpositions of coherent states, known as “cat states” may be exactly orthogonal to one another and used to form a computational basis.

[0007]Different types of superconducting qubits using Josephson junctions have been proposed, including “phase qubits,” where the computational basis is the quantized energy states of Cooper pairs in a Josephson Junction; “flux qubits,” where the computational basis is the direction of circulating current flow in a superconducting loop; and “charge qubits,” where the computational basis is the presence or absence of a Cooper pair on a superconducting island. Superconducting qubits are an advantageous choice of qubit because the coupling between two qubits is strong making two-qubit gates relatively simple to implement, and superconducting qubits are scalable because they are mesoscopic components that may be formed using conventional electronic circuitry techniques. Additionally, superconducting qubits exhibit excellent quantum coherence and a strong non-linearity associated with the Josephson effect. All superconducting qubit designs use at least one Josephson junction as a non-linear non-dissipative element.

BRIEF SUMMARY

[0008]According to some aspects, a quantum information processing (QIP) system is provided. The QIP system includes a data qubit and an ancilla qubit, the ancilla qubit having an asymmetric error channel. The data qubit is coupled to the ancilla qubit.

[0009]According to some aspects, a method of performing QIP in a system comprising a data qubit coupled to an ancilla qubit is provided. The method includes driving the ancilla qubit with a stabilizing microwave field to create an asymmetric error channel.

[0010]The foregoing is a non-limiting summary of the invention, which is defined by the appended claims.

BRIEF DESCRIPTION OF DRAWINGS

[0011]Various aspects and embodiments are described with reference to the following drawings. The drawings are not necessarily drawn to scale. For the purposes of clarity, not every component may be labeled in every drawing. In the drawings:

[0012]FIG. 1 is block diagram of a quantum information processing system, according to some embodiments.

[0013]FIG. 2 is a diagram of the joint system of the data qubit and the ancilla qubit of FIG. 1, according to some embodiments.

[0014]FIG. 3 is a diagram of a superconducting circuit element of FIG. 2 that includes a transmon, according to some embodiments.

[0015]FIG. 4 is a diagram of a superconducting circuit element of FIG. 2 that includes a superconducting nonlinear asymmetric inductor element (SNAIL), according to some embodiments.

[0016]FIG. 5 is a block diagram of a quantum information system based on cavity quantum electrodynamics, according to some embodiments.

[0017]FIG. 6 depicts a Bloch sphere based on cat states, according to some embodiments.

[0018]FIG. 7A depicts an eigenspectrum of a pumped cat oscillator, according to some embodiments.

[0019]FIG. 7B depicts the potential of a pumped cat oscillator (PCO) in the limit of large parametric drive, according to some embodiments.

[0020]FIG. 8 is a quantum circuit diagram of error syndrome detection, according to some embodiments.

[0021]FIG. 9A is a plot of the dynamics of a PCO and associated qubits during a stabilizer measurement for a four-qubit toric code, according to some embodiments.

[0022]FIG. 9B is a plot of the dynamics of a PCO and associated qubits during a stabilizer measurement for a four-qubit toric code, according to some embodiments.

[0023]FIG. 10A is a plot of the dynamics of a PCO and associated qubits during a stabilizer measurement for a cat code, according to some embodiments.

[0024]FIG. 10B is a plot of the dynamics of a PCO and associated qubits during a stabilizer measurement for a cat code, according to some embodiments.

[0025]FIG. 11A is a quantum circuit diagram for performing adaptive phase estimation, according to some embodiments.

[0026]FIG. 11B is a quantum circuit diagram for performing adaptive phase estimation, according to some embodiments.

[0027]FIG. 12 is a diagram illustrating a readout process in terms of a Bloch sphere, according to some embodiments.

[0028]FIG. 13 is a schematic of a quantum information processing system, according to some embodiments.

[0029]FIG. 14A is a plot of the amplitude as a function of time of multiple driving fields used to implement a bias-preserving CNOT gate, according to some embodiments.

[0030]FIG. 14B is a plot of the phases as a function of time of multiple driving fields used to implement a bias-preserving CNOT gate, according to some embodiments.

[0031]FIG. 15 is a quantum circuit diagram of a technique for detecting errors, according to some embodiments.

[0032]FIG. 16 is a flowchart of a quantum information processing method, according to some embodiments.

[0033]FIG. 17 is a flowchart of a readout method, according to some embodiments.

DETAILED DESCRIPTION

[0035]To perform useful a quantum information process, conventional quantum information systems initialize a set of qubits, referred to as “data qubits” because they are used to encode the information being processed, to a particular quantum state, implement a set of quantum gates on the qubits, and measure the final quantum state of the qubits after performing the quantum gates. A first type of conventional quantum gate is a single-qubit gate, which transforms the quantum state of a single qubit from a first quantum state to a second quantum state. Examples of single-qubit quantum gates include the set of rotations of the qubit on a Bloch sphere. A second type of conventional quantum gate is a two-qubit gate, which transforms the quantum state of a first qubit based on the quantum state of a second qubit. Examples of two-qubit gates include the controlled NOT (CNOT) gate and the controlled phase gate. Conventional single-qubit gates and two-qubit gates unitarily evolve the quantum state of the qubits from a first quantum state to a second quantum state.

[0036]For large-scale quantum computation to be viable, the quantum states used in the QIP must be protected from errors, which result from inevitable and uncontrolled interactions with the environment. Techniques for mitigating such errors include quantum error correction (QEC) schemes. In some conventional QEC schemes, quantum information is protected by linking errors and undesirable interactions with low-weight quantum operators. For example, quantum information may be encoded in a logical qubit using the non-local degrees of freedom of a high-dimensional system rather than simply encoding the information in a the two quantum states of a physical qubit. In such encodings, high-weight operators imply many-body operators arising, for example, in a system of several qubits or operators involving many quantum states of a single high-dimensional physical system (e.g., a quantum mechanical oscillator). The high-weight operators characterizing a codespace of quantum information are referred to as “stabilizers” and are designed to commute with the logical qubit operators but anti-commute with the errors in the system. In the absence of errors, the system lies in the +1 eigenspace of the stabilizer and after an error occurs the system moves to the −1 eigenspace. Consequently, the location and type of errors can be determined from the result of measuring the stabilizers, which are also known as an “error syndrome.” Measurement of these high-weight stabilizers uses highly-engineering, highly-unnatural, many-body interactions between components of the quantum system.

[0037]The inventors have recognized and appreciated that the above types of QEC techniques are undesirable for practical implementations of QIP. Instead, the inventors have recognized and appreciated that it is desirable to synthesize stabilizer measurements via naturally available couplings between the data qubits of the system and an ancillary system. Coupling the data qubits of the QIP system exposes the data qubits to a different set of errors that may be just as challenging to mitigate. For example, if the measurement of the ancillary system is not designed intelligently, errors from the ancillary system may propagate to the data qubits, damaging the encoded quantum information beyond repair. Recognizing this, the inventors have developed techniques for reducing and/or, in some instances, eliminating such catastrophic backaction from the ancillary system.

$\hat{V}={\hat{\sigma }}_z\sum _{i=1}^N{g}_i\left(t\right){\hat{M}}_i$

where σ_z is the z-Pauli operator of the ancilla qubit and g_i are controllable interaction strengths between the ancilla and each of the physical qubits used to form the logical data qubit. The evolution of the joint system of the data qubit and the ancilla qubit is described by the unitary operator:

$\begin{array}{c}\hat{U}\left(t\right)=𝒯\exp \left(-i{\int }_0^t\hat{V}\left(\tau \right)d\tau \right)\\ =\cos \left(\sum _{i=1}^N{\int }_0^t{g}_i\left(\tau \right){\hat{M}}_{i}d\tau \right)+i\sin \left(\sum _{i=1}^N{\int }_0^t{g}_i\left(\tau \right){\hat{M}}_{i}d\tau \right){\hat{\sigma }}_z.\end{array}$

[0039]The coupling strength and duration of the interaction, T, between the data qubit and the ancilla qubit may be chosen such that the unitary operator acting on the joint system (up to local rotations) becomes:

$\hat{U}\left(T\right)=\frac{1+\hat{S}}{2}+\frac{1-\hat{S}}{2}{\hat{\sigma }}_z.$

[0040]The result of the interaction with an interaction time T, is therefore a phase-flip of the ancilla qubit conditioned on whether the stabilizer is +1 or −1. This phase-flip in the ancilla qubit is the error syndrome.

[0041]The inventors have recognized and appreciated that, during the interaction time, the data qubit and the ancilla qubit are entangled and, to be a successful QEC scheme, it is desirable to engineer the joint system such that errors in the ancilla qubit do not propagate as uncorrectable errors to the data qubit, which is known as “fault-tolerance.” To prevent the propagation of uncorrectable errors to the data qubit and achieve fault-tolerance, all possible errors in the ancilla qubit should commute with the unitary operator Û(t) at all times. In the above example, the phase flip error {circumflex over (σ)}_Z satisfies this condition. Therefore, if a phase-flip error occurs at any time τ during the interaction time duration, then at time T, the state of the system is described by the unitary operator:

$\hat{U}\left(T-\tau \right){\hat{\sigma }}_z\hat{U}\left(\tau \right)={\hat{\sigma }}_{z}U\left(T\right)=\frac{{\hat{\sigma }}_z\left(1+\hat{S}\right)}{2}+\frac{1-\hat{S}}{2}.$

[0042]Based on this unitary operation, it is clear that the phase flip in the ancilla qubit only introduces an error in the measurement of the syndrome, but does not cause any backaction on the data qubit. Importantly, however, bit flip errors (represented by the Pauli matrix {circumflex over (σ)}_x and amplitude damping errors (represented by the Pauli matrix {circumflex over (σ)}₋) in the ancilla qubit do not commute with the unitary operator Û(t). In fact, a bit flip error, {circumflex over (σ)}_x on the ancilla qubit propagates as a high-weight error to the data qubit.

[0043]Conventional approaches to fault-tolerant extraction of error syndromes are based on using multiple ancillas prepared in complex quantum states, performing multiple bitwise entangling gates between the data qubits and the ancilla qubits, and then measuring the ancilla qubits. The inventors have recognized and appreciated that these conventional approaches lead to rapidly growing overhead of computationally expensive entangling gates and ancilla hardware, forcing more stringent requirements on error rates and making fault-tolerant quantum computation impractical, if not impossible, at a large scale. Moreover, the inventors have recognized and appreciated that an efficient fault-tolerant syndrome extraction scheme would enable large-scale quantum information processing. Accordingly, some aspects of the present application are directed to efficient fault-tolerant syndrome extraction.

[0044]The inventors have recognized and appreciated that, in the stabilizer measurement scheme described above, the unitary operator Û(t) would result in no backaction on the data qubit if the ancilla qubit did not have bit flip {circumflex over (σ)}_x errors. Thus, some aspects of the present application are directed to using an ancilla qubit with an asymmetric error channel where bit flip errors are suppressed relative to phase flip errors. By suppressing the bit-flip errors, which do not commute with the unitary operator Û(t), it is possible to engineer a physical unitary operation that nearly commutes with the ancilla's error channel and will therefore be effectively transparent to ancilla errors.

[0045]Aspects of the present disclosure include a method for making fault-tolerant measurements in quantum systems. The techniques described herein may be used in at least three possible applications. First, the techniques may be used in quantum error correction schemes by allowing fault-tolerant extraction of error syndromes. Second, the techniques may be used for new, more efficient error correcting codes and procedures. Third, the techniques may be used to create bias-preserving gates, such as a controller-NOT (CNOT) gate.

[0046]The inventors have recognized and appreciated that it is possible to perform a fault-tolerant extraction of an error syndrome using only local operations with an ancilla whose error channel is strongly biased (i.e., asymmetric). Some embodiments improve upon the overhead requirements of relative to conventional schemes fault-tolerant syndrome measurements. Some embodiments include a hardware efficient realization of such a syndrome extraction scheme using a two-component cat state in a parametrically driven nonlinear oscillator that exhibits a highly-biased noise channel.

[0047]The inventors have further recognized and appreciated the flexibility of the above approach. In some embodiments, different codes may be used. In some embodiments, the syndrome extraction process is used for a variety of codes such as qubit-based toric codes, bosonic cat-codes (and in extension, binomial and pair-cat code) and Gottesman-Kitaev-Preskill (GKP) codes. However, other codes may also be used.

[0048]A challenge for error correction with biased noise is to be able to maintain the bias while performing elementary gate operations such as a CNOT gate, which is an important ingredient for many error correction codes and for universal computation. In conventional systems that use physical qubits as data and/or ancilla qubits, a native bias-preserving CNOT is not possible even if the underlying noise is biased. The inventors have recognized and appreciated that the aforementioned techniques developed for fault-tolerant syndrome extraction can be utilized and extended to realize a bias-preserving CNOT gate between two stabilized cat states. In some embodiments, a CNOT gate is based on the structure of cat states in phase space. In this case, a stabilized cat state can be realized in a parametrically driven nonlinear cavity or via dissipation engineering. Some embodiments that include a bias preserving CNOT gate may achieve gains in the threshold for topological error correcting codes (e.g. toric and surface codes).

[0049]In some embodiments, when combined with an ZZ(θ) gate, it may be possible to reduce the thresholds for what is known as “magic state preparation” (which is an important but expensive ingredient, in terms of overhead costs, to achieve universality). In some embodiments, the ZZ(θ) gate inherently preserves bias and may be implemented with stabilized cats. The inventors have recognized and appreciated that combining bias preserving CNOT gates, ZZ(θ) gates and syndrome measurements provides the basis for a fault-tolerant architecture for large-scale quantum computation with ultrahigh thresholds and drastically reduced overhead requirements. Such an architecture, which exploits the bias in the noise channel of stabilized cat qubits, does not have any equivalent in conventional systems based on physical qubits.

[0050]FIG. 1 illustrates a QIP system according to some embodiments. The QIP system 100 includes at least a data qubit 110 and an ancilla qubit 120. Some embodiments further include a microwave field source 150 and/or a measurement device 125. The measurement device 125 may include a read-out cavity 130 and a cavity state detector 140. Though not illustrated as such, the microwave field source 150 may be considered to be part of the measurement device 125 as microwave fields emitted by the microwave field source 150 play a role in the measurement.

[0051]The data qubit 110 may be any physical or logical qubit capable of being coupled to the ancilla qubit 120. In some embodiments, the data qubit 110 may include a superconducting circuit component. For example, the data qubit 110 may include at least one Josephson junction. In some embodiments, the data qubit 110 may include a transmon. In some embodiments, the data qubit 110 may include a superconducting nonlinear asymmetric inductor element (SNAIL), which is an example of a superconducting circuit component that includes multiple Josephson Junctions. In other embodiments, the data qubit 110 may include an oscillator. An example of a linear oscillator that may be used includes the electromagnetic field, e.g., microwave radiation, supported by a cavity. A cavity may include a three-dimensional (3D) cavity or a planar transmission line cavity. In some embodiments, the cavity may be driven to include a specific type of quantum state. For example, as described in more detail below, the cavity may be driven to include a cat state or a GKP state. In some embodiments, a superconducting circuit component may be coupled to a cavity to form a Kerr-nonlinear cavity.

[0052]The ancilla qubit 120 may be any physical or logical qubit capable of being coupled to the data qubit 110. In some embodiments, the ancilla qubit 120 may include a superconducting circuit component. For example, the ancilla qubit 120 may include at least one Josephson junction. In some embodiments, the ancilla qubit 120 may include a transmon. In some embodiments, the ancilla qubit 120 may include a SNAIL. In other embodiments, the ancilla qubit 120 may include an oscillator. An example of a linear oscillator that may be used includes the electromagnetic field, e.g., microwave radiation, supported by a cavity. A cavity may include a three-dimensional cavity or a planar transmission line cavity. In some embodiments, the cavity may be driven to include a specific type of quantum state. For example, as described in more detail below, the cavity may be driven to include a cat state or a GKP state. In some embodiments, a superconducting circuit component may be coupled to a cavity to form a Kerr-nonlinear cavity.

[0053]The ancilla qubit 120 may be used by the measurement device 125 to measure one or more properties of the data qubit 110. For example, an interaction between the data qubit 110 and the ancilla qubit 120 may be engineered such that the state of the ancilla qubit 120 is based on a particular property of the data qubit 110. In some embodiments, the measurement of the data qubit 110 is a quantum nondemolition measurement, meaning the state of the data qubit 110 is left unaffected by the measurement process. In some embodiments, the quantum nondemolition measurement may be performed by using the measurement device 125 to measure the state of the ancilla qubit 120, after the data qubit 110 and the ancilla qubit 120 interact, to determine a property of the ancilla qubit 120. In some embodiments, the interaction between the data qubit 110 and the ancilla qubit 120 may be turned on by driving the data qubit 110 and/or the ancilla qubit 120 with one or more microwave fields using the microwave field source 150.

[0054]The read-out cavity 130 is a cavity coupled to the ancilla qubit 120 and configured to support multiple electromagnetic radiation, e.g., microwave radiation, states based on a property of the ancilla qubit 120. In some embodiments, an interaction between the read-out cavity 130 and the ancilla qubit 120 is engineered such that the state of the read-out cavity 130 is dependent on a particular property of the ancilla qubit 120, which itself may be based on a property of the data qubit 110. For example, if a property of the ancilla qubit 120 is a first value, then the interaction results in the read-out cavity 130 being in a first state; and if the property of the ancilla qubit 120 is a second value, then the interaction results in the read-out cavity being in a second state. In some embodiments, the two states of the read-out cavity 130 may be two different quasi-orthogonal coherent states. In other words, the read-out cavity 130 may be displaced in different ways depending on the value of the property of the ancilla qubit 120. In some embodiments, this process may be performed using what is referred to herein as a “Q-switch,” which uses a frequency conversion technique to conditionally displace the read-out cavity 130 based on the property of the ancilla qubit 130. In some embodiments, the interaction between the read-out cavity 130 and the ancilla qubit 120 may be turned on by driving the read out cavity 130 and/or the ancilla qubit 120 with one or more microwave fields using the microwave field source 150.

[0055]The cavity state detector 140 may be, for example a microwave radiation detector capable of distinguishing between the possible states of the read-out cavity 130 that result from the interaction between the read-out cavity 130 and the ancilla qubit 120. In some embodiments, the cavity state detector may be a phase-sensitive detector that is capable of measuring not only amplitude, but phase of the electromagnetic field of the read-out cavity 130. For example, the cavity state detector 140 may be a homodyne detector or a heterodyne detector. The result of the detection, in some embodiments, is directly related to the error syndrome.

[0056]In some embodiments, the ancilla qubit 130 includes a cavity, the state of which may be measured directly using a homodyne detector. However, if the cavity of the ancilla qubit 130 is a high-Q cavity, the homodyne detection would be slow. Accordingly, the read-out cavity may be a low-Q cavity that may be readout quickly.

[0057]FIG. 2 is a diagram of a particular embodiment of a joint system 200 that includes an example of the data qubit 110 and an example of the ancilla qubit 120, according to some embodiments. The data qubit 110 includes a data cavity 210 and a data superconducting circuit 212. The ancilla qubit 120 includes an ancilla cavity 220 and an ancilla superconducting circuit 222. The two cavities are coupled together with via an interface 230, which may include, for example, a microwave waveguide and/or a pin connector.

[0058]The data cavity 210 may be a three-dimensional cavity and includes at least one microwave port 214 for receiving microwave fields 216 from the microwave field source 150. The ancilla cavity 220 may be a three-dimensional cavity that includes at least one microwave port 224 for receiving microwave fields 226 from the microwave field source 150. In some embodiments, the microwave ports may include pin connectors and/or microwave waveguides. While FIG. 2 illustrates only a single port for each cavity, each cavity may include more than one port for receiving and/or transmitting microwave fields. For example, not shown in FIG. 2 is a port for coupling the ancilla cavity 220 to the read-out cavity 130.

[0059]In some embodiments, the data superconducting circuit element 212 and the ancilla superconducting circuit element 222 may include a nonlinear circuit element. For example, the superconducting circuit elements may be a transmon or a SNAIL. FIG. 3 illustrates an example of a superconducting circuit element 300 that may be used as the data superconducting circuit element 212 and/or the ancilla superconducting circuit element 222. The superconducting circuit element 300 includes a transmon 301 that consists of a single Josephson junction and an antenna that includes a first antenna portion 303 and a second antenna portion 305. The two antenna portions together form a dipole antenna through which the transmon 301 is coupled to the three-dimensional cavity in which the superconducting circuit element 300 is located.

[0060]FIG. 4 illustrates an example of a superconducting circuit element 400 that may be used as the data superconducting circuit element 212 and/or the ancilla superconducting circuit element 222. The superconducting circuit element 400 includes a SNAIL 401 that consists of a single Josephson junction and an antenna that includes a first antenna portion 403 and a second antenna portion 405. The two antenna portions together form a dipole antenna through which the SNAIL 401 is coupled to the three-dimensional cavity in which the superconducting circuit element 400 is located.

[0061]The SNAIL 401 is a nonlinear superconducting circuit element that has additional tenability relative to a transmon. FIG. 5 is a schematic diagram of a SNAIL 500, according to some embodiments. The SNAIL 500 includes a superconducting ring 501 with two nodes 511 and 512. There are two path along two different portions of the superconducting ring 501 that connect the first node 511 and the second node 512.

[0062]The first ring portion includes multiple Josephson junctions 505-507 connected in series. In some embodiments, there are no other circuit elements between one Josephson junction and the next Josephson junction. For example, a Josephson junction is a dipole circuit element (i.e., it has two nodes). A first node of a first Josephson junction 505 is directly connected to the first node 511 of the SNAIL, which may lead to some other external circuit element (not shown). A second node of the first Josephson junction 505 is directly connected to a first node of a second Josephson junction 506. A second node of the second Josephson junction 506 is directly connected to a first node of a third Josephson junction 507. A second node of the third Josephson junction 507 is directly connected to a second node 512 of the SNAIL, which may lead to some other external circuit element (not shown), such as a portion of an antenna.

[0063]While FIG. 5 illustrates the first ring portion including three Josephson junctions, any suitable number of Josephson junctions greater than one may be used. For example, three, four, five, six, or seven Josephson junctions may be used. Three Josephson junctions are selected for the example shown because three Josephson junctions is the lowest number of Josephson junctions (other than zero or one) that can be formed using a Dolan bridge process of manufacturing, which may be used in some embodiments.

[0064]In some embodiments, Josephson junctions 505-507 are formed to be identical. For example, the tunneling energies, the critical current, and the size of the Josephson junctions 505-507 are all the same.

[0065]The second ring portion of the SNAIL 500 includes a single Josephson junction 508. In some embodiments, there are no other circuit elements in the second ring portion. A first node of a single Josephson junction 508 is directly connected to the first node 511 of the SNAIL, which may lead to some other external circuit element (not shown), such as a portion of an antenna. A second node of the single Josephson junction 508 is directly connected to the second node 512 of the SNAIL, which may lead to some other external circuit element (not shown), such as a portion of an antenna.

[0066]The single Josephson junction 508 has a smaller tunneling energy than each of Josephson junctions 505-507. For this reason, the single Josephson junction 508 may be referred to as a “small” Josephson junction and Josephson junctions 505-507 may be referred to as “large” Josephson junctions. The terms “large” and “small” are relative terms that are merely used to label the relative size of Josephson junction 508 as compared to Josephson junctions 505-507. The Josephson energy and the Josephson junction size are larger in the large Josephson junction than in the small Josephson junction. The parameter a is introduced to represent the ratio of the small Josephson energy to the large Josephson energy. Thus, the Josephson energy of the large Josephson junctions 505-507 is E_j and the Josephson energy of the small Josephson junction 508 is αE_j, where 0>α<1.

[0067]The right side of FIG. 5 illustrates the circuit element symbol for the SNAIL 500, which is used in the superconducting circuit element 400 of FIG. 4. The parameters that characterize the SNAIL 500 are the Josephson energy E_j and the superconducting phase difference, φ, of the small Josephson junction 508. Of note is the fact that the SNAIL 500 has only two nodes 511 and 512, which may be connected to respective portions of an antenna.

[0068]While two separate 3D cavities, one for the data qubit 110 and one for the ancilla qubit 120, is illustrated in FIG. 2, other cavity arrangements may be used. In some embodiments, a the data superconducting circuit element 212 and the ancilla superconducting circuit element 222 may be located within a single, shared 3D cavity. In other embodiments, the data superconducting circuit element 212 and the ancilla superconducting circuit element 222 may be coupled to a respective two-dimensional (2D) transmission line cavity.

[0069]In either the embodiments using a transmon, as illustrated in FIG. 3, or a SNAIL, as illustrated in FIG. 4, the superconducting circuit element coupled to a cavity forms a Kerr-nonlinear oscillator, which may be used as the data qubit and/or the ancilla qubit. In some embodiments, a two-photon pump, received from the microwave field generator 150, may be used to create a two-component cat state:

[0071]Some embodiments extract an error syndrome based on conditional rotation of a cat state around the Z axis. This may be accomplished, in some examples, using only low-weight local interactions. In some embodiments, this fault-tolerant technique may be used with a variety of error correcting codes, such as stabilizer codes. Examples of stabilizer codes include, but are not limited to toric codes, bosonic cat codes, and GKP codes. Some embodiments may use non-stabilizer based error correcting codes, such as non-additive quantum codes. Additionally, some embodiments may use the asymmetric error channel of the ancilla qubit to perform fault-tolerant quantum gates.

[0072]In some embodiments, the interactions between a data qubit and an ancilla qubit are realized using the inherent nonlinearity of the ancilla qubit implemented using a cat state in a Kerr-nonlinear cavity. As such, some embodiments require no additional coupling elements. Thus, by exploiting the techniques described herein, hardware-efficient quantum information processing schemes can be realized.

[0073]Error Syndrome Detection

[0074]In some embodiments, a Kerr-nonlinear oscillator implemented, for example, using the hardware described above, may be driven by a two-photon drive with a frequency equal to twice a resonance frequency of the oscillator. When driven by such a microwave field, the oscillator is referred to as a pumped-cat oscillator (PCO) and the Hamiltonian in the rotating wave approximation is:

Ĥ_PCO=−Kâ^†2â²+P(â^†2+â²),

where â^† and â are the photon creation and annihilation operators of the PCO, K is the strength of the Kerr nonlinearity, and P is the strength of the two-photon drive field. Rewriting the PCO Hamiltonian in terms of the coherent state amplitude,

$\beta =\sqrt{\frac{P}{K}},$

results in:

Ĥ_PCO=−K(â^†2−β²)(â²−β²)+Kβ⁴.

[0076]In the rotating frame, the PCO Hamiltonian is described by quasi-energy eigenstates which exhibit negative energies. When a displacement transformation D(±β)=exp(±βâ^†∓βâ) is applied to the PCO Hamiltonian, the displaced Hamiltonian, Ĥ′, becomes:

Ĥ′=D(±β)Ĥ_PCOD^†(±β)=−4Kβ²â^†â−Kâ^†2â²∓2Kβ(â^†2â+h.c.),

where, because

$\beta =\sqrt{\frac{P}{K}},$

Ĥ_I=Σ_iχ_i(t){circumflex over (M)}_i(â^†+â).

$\hat{U}\left(T\right)=\frac{1+\hat{S}}{2}+\frac{1-\hat{S}}{2}.$

[0080]Based on the above, the ancilla cat state of the PCO undergoes a bit-flip conditioned on the stabilizer being Ŝ=+1 or Ŝ=−1. The error syndrome can be extracted, in some embodiments, by measuring the state of the PCO at time T.

[0081]In some embodiments, an alternative coupling of the form Σ_iχ_i(t)({circumflex over (L)}_i â^†+{circumflex over (L)}^†â), where {circumflex over (L)}^†+{circumflex over (L)}_i={circumflex over (M)}_i, may be used. For example, such a coupling may be used to extract the error syndrome when using the GKP code.

[0085]Error Channel Due to Single Photon Loss

$\langle {𝒞}_{\beta }^{-}\left|{\hat{a}}^{†}\hat{a}\right|{𝒞}_{\beta }^{-}\rangle =\frac{{\beta }^2}{p^2},$

[0088]Other Noise Sources

[0089]In some embodiments, there are other sources of noise, such as photon gain, pure-dephasing, two-photon loss. Single-photon gain and pure dephasing may result in leakage out of the cat state subspace. But leakage can suppressed by ensuring the spectral densities of these noise sources are narrower than the energy gap between the cat state subspace and the other states of the eigenspectrum. Accordingly, some embodiments are engineered such that the PCO has single-photon gain and pure dephasing spectral densities less than the energy gap. In such embodiments, irrespective of the underlying cause of noise, the PCO's error channel is dominated by phase-flip errors, while bit-flip errors are exponentially suppressed.

[0090]Further, in some embodiments, it is possible that spurious excitations or sudden non-perturbative effects overcome the energy barrier and cause excitations to the states outside of the cat state subspace. Remarkably, fault-tolerance of the syndrome measurements is still preserved under these errors.

[0093]Example Stabilizer Measurements: Toric Codes

[0095]In some embodiments, direct measurement of the stabilizer Ŝ_z would require a five-body interaction between the data qubits and an ancilla qubit, which is challenging to realize experimentally. Instead, some embodiments perform a syndrome measurement using only two-body interactions. This may be accomplished by replacing {circumflex over (M)}_i with {circumflex over (σ)}_z,i in the interaction Hamiltonian described above. The resulting interaction Hamiltonian is:

Ĥ_I=χ(t)Ŝ′_z(â^†+â),

where Ŝ_z′={circumflex over (σ)}_z,1+{circumflex over (σ)}_z,2+{circumflex over (σ)}_z,3+{circumflex over (σ)}_z,4, which has the form of a longitudinal qubit-oscillator coupling. For simplicity, all the interaction strengths are assumed to be equal, though it is not required for them to be equal. As long as the interaction strengths are known, the duration of interaction with each qubit can be adjusted to perform the syndrome measurement. An alternate approach is to keep the duration of interaction fixed, but use a pair of bit-flip driving field pulses for each qubit appropriately separated in time.

[0096]Following the analysis of the example stabilizer measurement technique described above, the unitary operator corresponding to this interaction Hamiltonian is:

${\int }_0^{T_z}\chi \left(\tau \right)d\tau =\frac{\pi }{8}\beta .$

More generally, for embodiments where the interaction strengths are different for each qubit, then the interaction time duration for the i-th qubit is T_i,z, where

${\int }_0^{T_{i,z}}{\chi }_i\left(\tau \right)d\tau =\frac{\pi }{8}\beta$

for each qubit. At the end of the interaction time duration, the unitary operator reduces to:

$\hat{U}\left(T_z\right)=e^{\frac{i\pi \widehat{S_z^{\prime }}}{4}}\left[\left(\frac{1+{\hat{S}}_x}{2}\right)+\left(\frac{1-{\hat{S}}_z}{2}\right)\right].$

$|{\psi }_o\rangle =\frac{1}{\sqrt{8}}\left(\sum _i{\hat{\sigma }}_{x,i}+\sum _{i,j,k}{\hat{\sigma }}_{x,i}{\hat{\sigma }}_{x,j}{\hat{\sigma }}_{x,k}\right)|0,0,0,0\rangle .$

[0101]FIGS. 9A and 9B are plots of the dynamics of the PCO and the qubits during a stabilizer measurement for the rate of single-photon loss, κ=0 (solid lines) and κ=K/200, (dotted lines), for the values P=4K (which is equivalent to β=2),

$\chi =\frac{\pi }{2}{\chi }_0\sin \left(\frac{\pi t}{T_Z}\right),$

$|{\psi }_e\rangle =\frac{1}{\sqrt{8}}\left(\sum _{i,j}{\hat{\sigma }}_{x,i}{\hat{\sigma }}_{x,j}+\hat{I}+{\hat{\sigma }}_{x,1}{\hat{\sigma }}_{x,2}{\hat{\sigma }}_{x,3}{\hat{\sigma }}_{x,4},\right)|0,0,0,0\rangle .$

[0106]Example Stabilizer Measurements: Cat Codes

[0107]In some embodiments, a cat code stabilizer, which is a type of bosonic error correcting code where the information is encoded in superpositions of coherent states, is used. The stabilizer for the cat code is the photon-number parity operator {circumflex over (P)}=e_s^iπâs†âs, where â_s^† and â_s are the photon creation and annihilation operators for a data qubit (sometimes referred to here as a storage qubit or storage cat or storage oscillator). When measured, the photon-number parity operator indicates whether a state of the data qubit has an even or an odd number of photons.

[0109]In some embodiments for performing cat syndrome measurements, a storage oscillator, which encodes the cat codeword, is dispersively coupled to an ancilla qubit. The dispersive coupling between the storage oscillator and the ancilla qubit may be used to map the parity of the storage cat, which is a property of the data qubit, onto the ancilla qubit. However, a random relaxation of the ancilla during the measurement induces a random phase rotation of the cat codeword, making this scheme non-fault tolerant. The inventors have recognized and appreciated that a fault-tolerant syndrome detection scheme can be engineered by replacing the operator {circumflex over (M)} in the interaction Hamiltonian, Ĥ_I, above with the photon number operator {circumflex over (n)}=â_s^†â_s. In some embodiments, the interaction Hamiltonian of the storage oscillator and ancilla PCO is then given by:

Ĥ_I=χ(t)â_s^†â_s(â^†+â).

[0110]This interaction is equivalent to a longitudinal interaction between the storage oscillator and the ancilla PCO. In some embodiments, this interaction can be created in a tunable manner.

[0111]The unitary operator corresponding to this interaction Hamiltonian is:

${\int }_0^{T_P}\chi \left(\tau \right)d\tau =\frac{\pi }{4}\beta .$

At the end of the interaction time duration, T_p the unitary operator reduces to:

$\overline{U}\left(T_P\right)=e^{\frac{i\pi {\hat{a}}_s^{†}{\hat{a}}_s}{2}}\left[\left(\frac{1+\hat{P}}{2}\right)+\left(\frac{1-\hat{P}}{2}\right)\right].$

[0114]FIGS. 10A and 10B are plots of the dynamics of the PCO and the storage cavity during a stabilizer measurement for the rate of single-photon loss, κ=0 (solid lines) and κ=K/200, (dotted lines), for the values P=4K (which is equivalent to β=2)

$\chi =\frac{\pi }{2}{\chi }_0\sin \left(\frac{\pi t}{T_P}\right),$

[0116]In order to highlight the fault-tolerance of the measurements using a PCO, the case in which the measurement is carried out with a conventional two-level physical qubit with the same relaxation rate γ=K/200. With the conventional physical, the backaction on the storage cavity increases by approximately two-orders of magnitude.

[0117]Example Stabilizer Measurements: Gottesman-Kitaev-Preskill (GKP) Codes

[0118]In some embodiments, a GKP code, which is a type of bosonic error correcting code designed to correct random displacement errors in phase space, is used. In some embodiments, the codewords for the GKP code are the simultaneous +1 eigenstates of the phase-space displacements Ŝ_q=exp(2i√{square root over (π)}{circumflex over (q)})=D(i√{square root over (2π)}) and Ŝ_p=exp(−2i√{square root over (π)}{circumflex over (p)})=D(√{square root over (2π)}) of the storage cavity, where {circumflex over (q)} and {circumflex over (p)} are the position and momentum operators, respectively, defined in terms of the photon annihilation and creation operators of the storage cavity as {circumflex over (q)}=(â_s^†+â_s)/√{square root over (2)} and {circumflex over (p)}=i(â_s^†−â_s)/√{square root over (2)}, and D(i√{square root over (2π)}) and D(√{square root over (2π)}) are displacement operators, where D(β)=exp(βâ_s^†−β*â_s).

[0120]In some embodiments, a simple approach to measure the eigenvalues exp(i2√{square root over (π)}u) and exp(i2√{square root over (π)}v) of Ŝ_q and Ŝ_p, respectively, is based on an adaptive phase-estimation protocol (APE). In such embodiments, displacement operations are repetitively performed on the storage cavity, the displacement operations being conditioned on the state of the ancilla qubit. Thus, some embodiments are directed to a fault-tolerant protocol for the APE of the stabilizers for a GKP code using a stabilized cat in a PCO.

[0121]In some embodiments, to achieve a controlled displacement for use in APE, the storage cavity is coupled to the PCO via a tunable single photon exchange interaction (also known as a beam splitter operation), defined by the Hamiltonian:

Ĥ_BS=Ĥ_PCO+g(t)â^†â_s+g*(t)ââ_s^†,

where g(t) is a dynamic coupling strength between the storage cavity and the PCO. In some embodiments, this tunable beam splitter operation may be realized using the three- or four-wave mixing capability of the PCO and external microwave drives of appropriate frequencies received from the microwave field generator 150. For small values of |g|, the beam splitter Hamiltonian can be approximated as

${\hat{H}}_{\mathrm{BS}}^{\prime }={\hat{H}}_{\mathrm{PCO}}+\beta \left(\frac{p+p^{-1}}{2}\right)\left(g\left(t\right){\hat{a}}_s+g^{*}\left(t\right){\hat{a}}_s^{†}\right)-i\beta \left(\frac{p-p^{-1}}{2}\right)\left(g\left(t\right){\hat{a}}_s+g^{*}\left(t\right){\hat{a}}_s^{†}\right).$

[0122]For large amplitude β, the second term of the Hamiltonian Ĥ′_BS becomes negligibly small and evolution under the Hamiltonian results in a controlled displacement along the position or momentum quadrature depending on the phase chosen for the coupling g(t). In this limit, when the phase and amplitude of the coupling g(t) are chosen so that g(t)=g*(t)=|g(t)| and β∫₀^T1|g(t)|dt=√{square root over (π/2)}, the unitary operator corresponding to the beam splitter interaction Hamiltonian above reduces to:

${\hat{U}}_1\left(T_1\right)=D\left(-i\sqrt{\frac{\pi }{2}}\right)\left[\left(\right)D\left(i\sqrt{2\pi }\right)+\left(\right)\right].$

[0123]The above unitary operator, Û₁(T₁), is the conditional displacement of the storage cavity for APE of Ŝ_q, according to some embodiments.

[0124]Similarly, when the phase and amplitude of the coupling g(t) are chosen so that g(t)=i|g(t)|, g*(t)=i|g(t)|, and β∫₀^T2|g(t)|dt=√{square root over (π/2)}, the unitary operator corresponding to the beam splitter interaction Hamiltonian above reduces to:

${\hat{U}}_2\left(T_2\right)=D\left(-\sqrt{\frac{\pi }{2}}\right)\left[\left(\right)D\left(\sqrt{2\pi }\right)+\left(\right)\right].$

[0125]The above unitary operator, Û₂ (T₂), is the conditional displacement of the storage cavity for APE of Ŝ_p, according to some embodiments.

[0127]The protocol 1100 for estimating Ŝ_q includes performing a first joint unitary operation 1110 on the data qubit 1102 and the ancilla qubit 1103 such that Û₁(T₁) is implemented. In some embodiments, the first joint unitary operation 1110 includes two separate actions. First, a displacement operation 1111 that implements the displacement

$D\left(-\sqrt{\frac{\pi }{2}}\right)$

on the data qubit 1103 is performed. Then, a conditional displacement operation 1113 that implements the displacement D(√{square root over (2π)}) on the data qubit 1103 based on the state of the ancilla qubit 1102.

[0128]The protocol 1100 for estimating Ŝ_q then includes a rotation operation 1120 performed on the ancilla qubit 1102 around the Z-axis by an angle φ. In some embodiments, the rotation operation 1120 is performed by driving the ancilla qubit 1102, which may be a PCO, with a microwave field. In some embodiments, the value of φ may be determined by a previous iteration of the protocol 1100 for estimating Ŝ_p.

[0131]The protocol 1150 for estimating Ŝ_p includes performing a second joint unitary operation 1160 on the data qubit 1102 and the ancilla qubit 1103 such that Û₂(T₂) is implemented. In some embodiments, the second joint unitary operation 1160 includes two separate actions. First, a displacement operation 1161 that implements the displacement

$D\left(-i\sqrt{\frac{\pi }{2}}\right)$

on the data qubit 1103 is performed. Then, a conditional displacement operation 1163 that implements the displacement D (i√{square root over (2π)}) on the data qubit 1103 based on the state of the ancilla qubit 1102.

[0132]The protocol 1100 for estimating Ŝ_p then includes a rotation operation 1170 performed on the ancilla qubit 1102 around the Z-axis by an angle ϕ. In some embodiments, the rotation operation 1170 is performed by driving the ancilla qubit 1102, which may be a PCO, with a microwave field. In some embodiments, the value of ϕ may be determined by a previous iteration of the protocol 1100 for estimating Ŝ_p.

[0136]Based on the above, in some embodiments, the APE protocols 1100 and 1150 may be iterated to estimate the stabilizer eigenvalues. As the number of iterations of phase estimation increases, the accuracy of the estimates of u, v also increases and, consequently, the uncertainty of the eigenvalues exp(2i√{square root over (π)}u) and exp(2i√{square root over (π)}v) decreases.

[0137]Ancilla Readout

[0138]In several of the embodiments above (e.g., see FIG. 8 and FIGS. 11A-11B), a readout operation is performed to measure the ancilla qubit. While the state of the ancilla qubit may be directly measured by, in the case the ancilla qubit is a PCO, by directly measuring the state of the cavity using homodyne detection, such a measurement would be slow due to the high Q of the PCO cavity. Thus, in some embodiments, a read-out cavity with a Q-value smaller than the Q-value of the ancilla cavity is measured using homodyne detection after mapping the state of the ancilla qubit onto the state of the read-out cavity. In some embodiments, the readout of the PCO may be a quantum nondemolition (QND) measurement, though it need not be (e.g., it may be that the readout introduces bit-flips or other errors in the state of the ancilla qubit). Such non-QND measurements are possible because the interaction between the ancilla PCO and the data qubit may be turned off while the PCO is being measured such that ancilla errors do not propagate to the data qubit. Such direct measurements of the ancilla qubit may be performed using a superconducting transmon.

[0139]In some embodiments, the readout of the ancilla PCO includes a measurement along the Z-axis of the Bloch sphere and does not introduce any additional nonlinearities into the system. As discussed above, the states along the Z-axis of the Bloch sphere are approximately coherent states and may be measured using homodyne detection of the field at the output of the PCO. To overcome the slow speed of direct homodyne detection of the PCO cavity, a Q-switch operation is performed whereby the PCO stats is switched via frequency conversion into a low-Q read-out cavity. In some embodiments, the Q-switch operation conditionally displaces the readout cavity based on the state of the PCO along the Z-axis.

[0140]As discussed above, the read-out operation may include a first operation where the cat states of the PCO are rotated into coherent states. Then, the coherent state of the PCO is Q-switched into the readout cavity. Finally the readout cavity is measured. FIG. 12 illustrates the read-out process 1200 in terms of the Bloch sphere.

Ĥ=ε(â^†+â)−Kâ^†2â²+P(â^†2+â²)

[0142]After the PCO is transformed from cat states into coherent states via the above rotations, the states of the PCO lies along the Z-axis of the Bloch sphere. In some embodiments, the PCO is then coupled to an off-resonance readout cavity. In the absence of an external microwave drive field, the coupling between the PCO and the readout cavity is negligible due to a large detuning between the two. In some embodiments, a single-photon exchange coupling (a beam splitter coupling) is turned on by applying at least one microwave drive field from the microwave field generator to compensate for the frequency difference between the PCO and the readout cavity. A three- or four-wave mixing between the drives, the PCO and the readout cavity results in an interaction between the PCO and the readout cavity causing a resonant single photon exchange between the two. This controllable coupling is referred to a Q-switch. The result of the Q-switch operation is to displace the readout cavity conditions on the state of the PCO, as shown in phase space diagram 1207 of the read-out process 1200. The Q-switch Hamiltonian for this process is given by Ĥ_Q=g(â^†â_r+ââ_r^†), where â_r^† and â_r are the creation and annihilation operators of the readout cavity and g is the tunable coupling strength between the PCO and the readout cavity. For small values of g, the Q-switch Hamiltonian may be approximated as:

${\hat{H}}_Q=\mathrm{ℊ\beta }\left(\frac{p+p^{-1}}{2}\right)\left({\hat{a}}_r+{{\hat{a}}_r}^{†}\right)-i\mathrm{ℊ\beta }\left(\frac{p-p^{-1}}{1}\right)\left({\hat{a}}_r-{{\hat{a}}_r}^{†}\right).$

[0143]In cases of moderately large β, final term becomes negligibly small and the result is a displacement of the readout oscillator conditioned on the state of the PCO, where the amplitude of the readout cavity's field is

$\langle {\hat{a}}_r\rangle =\mp \frac{2ℊi\beta }{{\kappa }_r}\left(1-e^{-{\kappa }_{r}t/2}\right)$

where κ_r is the linewidth of the field.

[0144]After the readout cavity is conditionally displaced, a homodyne detector is used to determine the state of the readout cavity and, thereby, determine the state of the PCO, which is equivalent to extracting an error syndrome.

[0145]Bias-Preserving Quantum Gates

[0146]The inventors have recognized and appreciated that the above techniques of using an asymmetric error channel of an ancilla qubit to detect error syndromes may be extended to implement a bias-preserving quantum gate. For qubits with biased noise channels (i.e., asymmetric error channels), operations that do not commute with the dominant error type can un-bias, or depolarize, the noise channel of the qubit, thereby reducing the benefits of the biased noise channel.

[0147]To understand how non-commuting operations can un-bias the noise channel of a qubit with a biased noise channel, consider a system that preserves the noise bias. For example, consider the following two-qubit gate:

ZZ(θ)=exp[iθ{circumflex over (Z)}₁{circumflex over (Z)}₂/2]

where {circumflex over (Z)}_i is the Z-Pauli operator for the i-th qubit and θ is a tunable phase angle. When θ=π/2, the ZZ(θ) gate becomes a controlled-phase gate, also referred to as a CZ gate, up to local Pauli rotations and an overall phase. The ZZ(θ) gate may be implemented with an interaction Hamiltonian of the form Ĥ_ZZ=−V{circumflex over (Z)}₁{circumflex over (Z)}₂ with the unitary evolution given by Û_ZZ(t)=exp(iVt{circumflex over (Z)}₁{circumflex over (Z)}₂). Under this unitary evolution, the ZZ(θ) gate is realized after an interaction time duration T=θ/2V. If a phase-flip occurs in either one of the two qubits at a time r during the interaction time, the evolution is modified as follows:

Û_e(T)=Û(T−τ){circumflex over (Z)}_1/2Û(τ)={circumflex over (Z)}_1/2Û(T)

Thus, the erroneous gate operation is equivalent to an error-free gate followed by a phase flip. Accordingly, the ZZ(θ) gate preserves the error bias of the qubit.

[0148]On the other hand, the controlled NOT (CNOT) gate (also referred to as a CX gate) between two qubits may be implemented using the following CX Hamiltonian:

${\hat{H}}_{\mathrm{cx}}=V\left[\left(\frac{{\hat{I}}_1+{\hat{Z}}_1}{2}\right)\otimes {\hat{I}}_2+\left(\frac{{\hat{I}}_1+{\hat{Z}}_1}{2}\right)\otimes {\hat{X}}_2\right]$

with the unitary evolution given by Û_CX(t)=exp(iĤ_CXt), where the control qubit and target qubit of the CNOT gate are labelled by 1 and 2, respectively. Under this evolution, the CNOT gate is realized after an interaction time duration T=π/2V, such that

$U\left(T\right)=\left[\left(\frac{{\hat{I}}_1+{\hat{Z}}_1}{2}\right)\otimes |{\hat{I}}_2+\left(\frac{{\hat{I}}_1+{\hat{Z}}_1}{2}\right)\otimes {\hat{X}}_2\right],$

where an overall phase is ignored. In the case of this CNOT gate, a phase-flip error in the target qubit at time r during the interaction time modifies the unitary evolution as follows:

$\begin{array}{c}{\hat{U}}_e\left(T\right)=\hat{U}\left(T-\tau \right){\hat{I}}_1\otimes {\hat{Z}}_2\hat{U}\left(\tau \right)\\ ={\hat{I}}_1\otimes {\hat{Z}}_2{e}^{\mathrm{iV}\left(T-\tau \right)\left({\hat{I}}_1-{\hat{Z}}_1\right)\otimes {\hat{X}}_2}\hat{U}\left(T\right).\end{array}\hspace{1em}$

Thus, the phase-flip error in the target qubit introduces a phase-flip error in the control qubit, depending on when the phase error in the target qubit occurs. Importantly, the phase-flip of the target qubit during the CNOT gate propagates as a combination of a phase-flip error and a bit-flip error in the same qubit. Consequently, the CNOT gate reduces the bias of the noise channel by introducing bit-flips in the target qubit. Similarly, coherent errors in the gate operation that arise from uncertainty in V and T also give rise to bit-flip errors in the target qubit. As a consequence, a native bias-preserving CNOT gate is not possible to implement.

[0149]The inventors have recognized and appreciated that in the absence of a bias-preserving CNOT gate, alternate circuits are required to extract an error syndrome. These alternate circuits add complexity and limit the gains in fault-tolerance thresholds for error correction that result from using qubits with biased noise. The inventors have therefore developed a novel solution to this problem by engineering a bias-preserving CNOT gate using the same two-component cat states realized in a parametrically driven nonlinear oscillator described above.

[0150]As shown in FIG. 6 above, the Bloch sphere is oriented such that the superposition states are oriented along the Z-axis of the Bloch sphere. Moreover, for the purposes of the CNOT gate, the Z-axis is selected as the computational basis such that:

$0\rangle =\frac{C_{\alpha }^{+}\rangle +C_{\alpha }^{-}\rangle }{\sqrt{2}},1\rangle =\frac{C_{\alpha }^{+}\rangle -C_{\alpha }^{-}\rangle }{\sqrt{2}},$

where α is the complex amplitude of the coherent state associated with the cat states.

$\begin{array}{c}{\hat{H}}_0\left(\varphi \right)=-K{\hat{a}}^{\mathrm{†2}}{\hat{a}}^2+P\left({\hat{a}}^{\mathrm{†2}}{e}^{2i\varphi }+{\hat{a}}^2{e}^{-2i\varphi }\right),\\ =K\left({\hat{a}}^{\mathrm{†2}}-{\alpha }^2{e}^{-2i\varphi }\right)\left({\hat{a}}^2-{\alpha }^2{e}^{2i\varphi }\right)+\frac{P^2}{K}.\end{array}\hspace{1em}$

[0153]In some embodiments, a two-qubit bias-preserving CNOT gate is based on a conditional phase-space rotation of a target qubit based on the state of the control qubit. To show how the conditional rotation results in a CNOT gate, consider two PCOs, each stabilized/pumped with its own two-photon microwave pump field. The initial state of the two qubit system is:

$\begin{array}{c}\psi \left(0\right)\rangle =\left(c_00\rangle +c_11\rangle \right)\otimes \left(d_00\rangle +d_11\rangle \right)\\ =\left(c_00\rangle +c_11\rangle \right)\otimes \left[\left(d_0+d_1\right)C_{\alpha }^{+}\rangle +\left(d_0-d_1\right)C_{\alpha }^{-1}\rangle \right],\end{array}\hspace{1em}$

where the first and second terms in the tensor product refer to the control and target qubits, respectively, and the terms c_i and d_i are simply the probability amplitudes for each of the components of the superposition and can be arbitrarily chosen to be any initial state. If the phase of the two-photon drive applied to the target PCO is conditioned on the state of the control PCO, then the state of the system evolves as follows such that any given time t, the state is:

$\psi \left(t\right)\rangle =c_00\rangle \otimes \left[\left(d_0+d_1\right)C_{\alpha }^{+}\rangle +\left(d_0-d_1\right)C_{\alpha }^{-}\rangle \right]+c_11\rangle \otimes \left[\left(d_0+d_1\right)C_{\alpha e^{i\varphi \left(t\right)}}^{+}\rangle +\left(d_0-d_1\right)C_{\alpha e^{i\varphi \left(t\right)}}^{-}\rangle \right].$

[0154]If the time-varying phase, ϕ(t) is such that ϕ(0)=0 and ϕ(T)=π, then at time T, the state becomes:

$\begin{array}{c}\psi \left(T\right)\rangle =c_00\rangle \otimes \left\{\left(d_0+d_1\right)C_{\alpha }^{+}\rangle +\left(d_0-d_1\right)C_{\alpha }^{-}\rangle \right\}+\\ c_11\rangle \otimes \left\{\left(d_0+d_1\right)C_{\alpha e^{i\pi }}^{+}\rangle +\left(d_0-d_1\right)C_{\alpha e^{i\pi }}^{-}\rangle \right\}\\ =c_00\rangle \otimes \left\{\left(d_0+d_1\right)C_{\alpha }^{+}\rangle +\left(d_0-d_1\right)C_{\alpha }^{-}\rangle \right\}+\\ c_11\rangle \otimes \left\{\left(d_0+d_1\right)C_{\alpha }^{+}\rangle -\left(d_0-d_1\right)C_{\alpha }^{-}\rangle \right\}\\ =c_00\rangle \otimes \left(d_00\rangle +d_11\rangle \right)+c_11\rangle \otimes \left(d_01\rangle +d_10\rangle \right)\\ ={\hat{U}}_{\mathrm{CX}}\psi \left(0\right)\rangle .\end{array}\hspace{1em}$

[0156]It can be shown that, unlike the previously described CNOT gate using the {circumflex over (X)} operator, the CNOT gate based on topological phase described above preserves the bias in the noise channel of the qubits. In particular, a phase-flip error occurs in the control PCO during the CNOT gate evolution is equivalent to a phase-flip occurring on the control qubit after an ideal CNOT gate is performed. Similarly, a phase-flip error on the target PCO during the CNOT gate evolution is equivalent to phase-flip errors on the control and target qubits occurring after an ideal CNOT gate. Therefore, the CNOT gate according to some embodiments, does not un-bias the noise channel. This result contrasts with the aforementioned CNOT gate implements between two strictly two-level qubits and shows one advantage of using a larger Hilbert space (e.g., an oscillator) to perform quantum information processing.

${\hat{H}}_{\mathrm{CX}}\equiv -K\left({\hat{a}}_c^{\mathrm{†2}}-{\beta }^2\right)\left({\hat{a}}_c^2-{\beta }^2\right)-K\left[{\hat{a}}_t^{\mathrm{†2}}-{\alpha }^2{e}^{-2i\varphi \left(t\right)}\left(\frac{\beta -{\hat{a}}_c^{†}}{2\beta }\right)-{\alpha }^2\left(\frac{\beta -{\hat{a}}_c^{†}}{2\beta }\right)\right]\times \left[{\hat{a}}_t^{\mathrm{†2}}-{\alpha }^2{e}^{-2i\varphi \left(t\right)}\left(\frac{\beta -{\hat{a}}_c}{2\beta }\right)-{\alpha }^2\left(\frac{\beta -{\hat{a}}_c}{2\beta }\right)\right]-\frac{\varphi \left(t\right)}{4\beta }{\hat{a}}_t^{†}{\hat{a}}_t\left(2\beta -{\hat{a}}_c^{†}-{\hat{a}}_c\right).$

${\hat{H}}_{\mathrm{CX}}^{{0\rangle }_c}\equiv -K\left({\hat{a}}_c^{\mathrm{†2}}-{\beta }^2\right)\left({\hat{a}}_c^2-{\beta }^2\right)-K\left({\hat{a}}_t^{\mathrm{†2}}-{\alpha }^2\right)\left({\hat{a}}_t^2-{\alpha }^2\right).$

${\hat{H}}_{\mathrm{CX}}^{{1\rangle }_c}\equiv -K\left({\hat{a}}_2^{\mathrm{†2}}-{\beta }^2\right)\left({\hat{a}}_c^2-{\beta }^2\right)-K\left({\hat{a}}_t^{\mathrm{†2}}-{\alpha }^2{e}^{-2i\varphi \left(t\right)}\right)\left({\hat{a}}_t^2-{\alpha }^2{e}^{2i\varphi \left(t\right)}\right)-\stackrel{.}{\varphi }\left(t\right){\hat{a}}_t^{†}{\hat{a}}_t.$

$r=\frac{N_{+}}{N_{-}}\sim 1-e^{-2{\alpha }^2}.$

${\Phi }_g^{-}\left(t\right)-{\Phi }_g^{+}\left(t\right)=\frac{4\varphi \left(t\right){\alpha }^2{e}^{-2{\alpha }^2}}{1-e^{-4{\alpha }^2}}.$

This term leads to a dynamic phase that exactly cancels out the geometric phase. Thus, the CNOT Hamiltonian results in a two-qubit evolution that implements a bias-preserving CNOT gate.

[0160]In some embodiments, the physical realization of the bias-preserving CNOT gate using three-wave mixing between two oscillators. The natural coupling between two oscillators is a beam splitter coupling. Thus, in some embodiments, the oscillators are themselves fourth-order, Kerr nonlinear. As such, the three-wave mixing can be generated by parametrically driving the target oscillator at a frequency ω_d=2ω_t−ω_c, where ω_t and ω_c are the frequencies of the target and control oxillators, respectively. Under such a driving field, the fourth order nonlinearity converts a photon in the drive field and a photon in the control oscillator to two photons in the target oscillator. Thereby, an effective three-wave mixing is realized between the control and target oscillators. In some embodiments, the Kerr nonlinearity of the oscillators themselves is sufficient to realize the CNOT interaction Hamiltonian and no additional coupling elements are necessary. Moreover, because of the parametric nature of the interaction, the coupling is controllable.

[0161]FIG. 13 is a schematic of a quantum information processing device 1300 configured to implement a bias-preserving CNOT gate, according to some embodiments. FIG. 13 provides additional detail about the driving fields than are provided in, e.g., the block diagrams of FIG. 1 and FIG. 2. The schematic of FIG. 13 is a circuit diagram equivalent of the quantum information processing device 1300. The physical system, in some embodiments, is implemented as discussed in connection with FIGS. 1-5 above.

[0162]The quantum information processing device 1300 includes a control qubit 1301 and a target qubit 1303. In some embodiments, the qubits 1301 and 1303 are a Kerr nonlinear cavity. The nonlinearity of the cavity may be controlled using a superconducting circuit element, such as a transmon or a SNAIL, as described above. In the example shown in FIG. 13, both the control qubit 1301 and the target qubit 1303 include a SNAIL. The SNAIL of the control qubit 1301 has a resonance frequency of co, and the SNAIL of the target qubit 1303 has a resonance frequency of ω_t. In some embodiments, the SNAILs are biased with an external magnetic field to engineer three- and/or four-wave mixing interactions between the control qubit 1301 and the target qubit 1303. Using this engineered interaction, a two-photon driven Kerr nonlinear oscillator results and may be used to create a PCO with a biased noise channel.

[0163]The control qubit 1301 and the target qubit 1303 are capacitively coupled to one another, as illustrated by the capacitor 1309. Microwave fields may be coupled to the control qubit 1301 via an input port 1305 and microwave fields may be coupled to the target qubit 1303 via an input port 1307. Microwave fields may be received from the microwave field generator 150, discussed in connection with FIG. 1. In some embodiments, microwave fields of more than one frequency may be applied to a given input port at one time.

[0164]To conditionally rotate the state of the target PCO in phase space based on the state of the control PCO, the CNOT Hamiltonian Ĥ_CX described above is implemented. Expanding the terms of the CNOT Hamiltonian can help understand what driving fields are needed to implement this Hamiltonian. The expanded CNOT Hamiltonian may be written as:

Ĥ_CX=−Kâ_c^†2â_c²−Kâ_t^†2â_t²+β²(â_c^†2+h.c.)+Kα² cos(ϕ(t))(e^iϕ(t)â_t^†2+h.c.)−(Kα² sin(ϕ(t))/(β)(iâ_t^†2â_c+h.c.)+(Kα⁴/2β)sin(2ϕ(t))(iâ_c^†+h.c.)−(Kα⁴ sin²(ϕ(t))/β²)â_c^†â_c−{dot over (ϕ)}(t)â_t^†â_t/2+({dot over (ϕ)}(t)/4β))â_t^†â_t(â_c^†+h.c.)

[0165]This expression can be further simplified by transforming the Hamiltonian into a rotating frame in which the frequencies of the two PCOs are both zero as:

{circumflex over (H)}′_CX=−Kâ_c^†2â_c²−Kâ_t^†2â_t²+β²(â_c^†2e^−2iθ(t)+h.c.)+Kα² cos(2ϕ(t))(â_t^†2+h.c.)−(Kα²/β)sin(ϕ(t))(iâ_t^†2â_ce^{iθ(t)−ϕ(t)}+h.c.)+(Kα⁴/2β)sin(2ϕ(t))(iâ_c^†e^−iθ+h.c.)+({dot over (ϕ)}(t)/4β))â_t^†â_t(â_c^†e^−θ(t)+h.c.)

where

$\theta \left(t\right)={\int }_0^t\left(\frac{K{\alpha }^4}{{\beta }^2}\right){\sin }^2\left(\varphi \left(s\right)\right)\mathrm{ds}.$

In this form, it becomes clear how the Hamiltonian can be parametrically engineered using for-wave mixing based on the Kerr-nonlinearity and driving fields. For example, in embodiments where a PCO is realized using a SNAIL, the terms proportional to â_i^†2 and â_i², where i=c, t labels the control (c) and target (t) qubits, are realized using three wave mixing and the terms proportional to â_t^†2â_c, â_c^†2â_t, â_t^†â_tâ_c^†, and â_t^†â_tâ_c are realized using four-wave mixing. Terms proportional to â_c and â_c^† do not require a nonlinearity and are realized by simply applying a drive field to the control qubit. Additionally, ϕ(t) is a phase shift that changes over time and adiabatically increases from 0 to π in the time T. Using all of the above information, the CNOT Hamiltonian can be expressed in terms of microwave field amplitudes and phases as:

{circumflex over (H)}′_CX=−Kâ_c^†2â_c²−Kâ_t^†2â_t²+A₁(â_c^†2e^iΦ1(t)+h.c.)+A₂(â_t^†2e^iΦ2(t)+h.c.)+A₃(â_t²â_ce^iΦ3+h.c.)+A₄(â_c^†e^iΦ4(t)+h.c.)+A₅â_t^†â_t(â_c^†e^iΦ5+h.c.),

where the field amplitudes, A_i, are assumed to be positive. The driving microwave fields corresponding to the amplitudes A₁, A₂, A₃, A₄, and A₅ are applied at the frequencies 2ω_c, 2ω_t, 2ω_t−ω_c, ω_c, and ω_c, respectively.

[0166]In some embodiments, a particular sequence of fields are applied to the control qubit 1301 and/or the target qubit 1303 during an interaction time duration T. It is during this interaction time that the execution of the CNOT gate is performed. In some embodiments, during the times outside of this interaction time duration, the amplitudes and phases take on the following fixed values: A₁=Kβ², Φ₁=0, A₂=Kα², Φ₂=0, A₃=A₄=A₅=Φ₃=Φ₄=Φ₅=0. During the CNOT interaction time, the phases Φ_i (t) are time-varying and change from a value of 0 to π. The value of the phases between 0 and T may change in any way, as long as the changes are adiabatic. In some embodiments, the phases change linearly. For example, Φ_i(t)=πt/T.

[0167]FIG. 14A is a plot of the amplitude as a function of time of the five driving fields used to implement the CNOT gate, according to some embodiments, and FIG. 14B is a plot of the time-dependent phases as a function of time for the five driving fields used to implement the CNOT gate.

[0168]First, a first microwave field is applied to the control cavity at a frequency 2ω_c with a fixed amplitude A₁ and a time-dependent phase Φ₁(t). This first microwave field provide the two-photon term to drive the control cavity via three-wave mixing. The fixed amplitude A₁ is illustrated by line 1401 in FIG. 14A and the time-dependent phase Φ₁(t) is illustrated by line 1411 in FIG. 14B. In the example shown, the phase decreases linearly.

[0169]Next, a second microwave field is applied to the target cavity at a frequency 2ω_t with a time-dependent amplitude A₂ and a time dependent-phase ∠₂(t). This second microwave field provide the two-photon term to drive the target cavity via three-wave mixing. The changing amplitude A₂ is illustrated by line 1402 in FIG. 14A and the time-dependent phase Φ₂ (t) is illustrated by line 1412 in FIG. 14B. The amplitude changes sinusoidally over time. In the example shown, the phase is constant at a first phase value during a first portion of the gate time duration and constant at a second phase value during a second portion of the gate time duration, wherein the first phase value is less than the second phase value. This is because the amplitudes are always taken to be positive. When the amplitude A₂, which is a sine function, crosses the zero amplitude point, rather than going negative, the amplitude begins to increase again, and the phase takes on different value instead.

[0170]A third microwave field at a frequency 2ω_t−ω_c is applied to the target cavity at with a time-dependent amplitude A₃ and a time-dependent phase Φ₃(t). This third microwave field realize the coupling terms proportional to â_t^†2â_c inn the CNOT Hamiltonian. The changing amplitude A₃ is illustrated by line 1403 in FIG. 14A and the time-dependent phase Φ₃(t) is illustrated by line 1413 in FIG. 14B. The amplitude changes as a cosine over time. In the example shown, the phase increases linearly as a function of time.

[0171]A fourth microwave field at a frequency ω_c is applied to the control cavity at a with a time-dependent amplitude A₄ and a time-dependent phase Φ₄(t). This fourth microwave field realizes the single-photon drive of the control cavity. The changing amplitude A₄ is illustrated by line 1404 in FIG. 14A and the time-dependent phase Φ₄(t) is illustrated by line 1414 in FIG. 14B. The amplitude changes as a cosine over time. In the example shown, the phase decreases linearly during the first portion of the gate time duration and decreases linearly during the second portion of the gate time duration. The linear decrease has the same slope in both portions of the gate time duration, but there is a jump in the phase half way through the gate time duration. This is because the amplitudes are always taken to be positive. When the amplitude A₄, which is a cosine, crosses the zero amplitude point, rather than going negative the amplitude begins to increase again and the phase jumps to a different value instead.

[0172]Finally, a fifth microwave field is applied to the target cavity at a frequency co, with a fixed amplitude A₅ and a time-dependent phase Φ₅(t). This fifth microwave field provide realizes the final term in the CNOT Hamiltonian. The fixed amplitude A₅ is illustrated by line 1405 in FIG. 14A and the time-dependent phase Φ₅(t) is illustrated by line 1415 in FIG. 14B. In the example shown, the phase decreases linearly.

[0173]Error-Correction Code Tailored to Biased Noise

[0176]The n−1 stabilizer generators for the repetition code are {circumflex over (X)}₁ ⊗{circumflex over (X)}₂ ⊗Î₃ ⊗Î₄ . . . , Î₁ε{circumflex over (X)}₂⊗{circumflex over (X)}₃⊗Î₄ . . . , etc. In some embodiments, the most naive way to detect errors is used, which is to measure each stabilizer generator using an ancilla. Such a technique is shown by the quantum circuit diagram 1500 of FIG. 15. The horizontal lines 1501-1503 represent the code qubits and the horizontal lines 1504-1505 represent the ancilla qubits in an example where n=3.

[0178]This decoding scheme is equivalent to constructing an r-bit repetition code for each of the (n−1) stabilizer generators of the repetition code. Thus, each bit of syndrome from the inner code is itself encoded in an [r, 1, r] repetition code so that decoding can proceed by first decoding the syndrome bits and then decoding the resulting syndrome. This naive way to decode the syndrome results in a simple analytic expressions for the logical error rates. However, the inventors have recognized and appreciated that this naive may not be the preferred approach to decode and, in some embodiments, the two-stage decoder of FIG. 15 can be replaced by a decoder that directly infers the most likely error on the n-qubit repetition code given s measured syndrome bits. Accordingly, in some embodiments, the notion of a measurement code is introduced that exploits the above insights to improve on the naive scheme by constructing a block code that can directly correct the bit-flip errors on the n data qubits in a single decoding step.

[0179]In some embodiments, to construct a measurement code the syndrome measurement procedure measures a total of s elements of the stabilizer group (not necessarily the specified generators) by coupling to ancilla qubits and corrects any t=(d−1)/2 phase-flip errors on the n qubits. Thus, there is a classical code with parameters [n+2, n, d]. However, not every classical code with those parameters is admissible, because the classical parity checks should be compatible with the stabilizers of the original quantum code, in this example the repetition code. In particular, each parity check in the measurement code should have even weight when restricted to the data qubits so that it commutes with the logical {circumflex over (Z)}_L operator of the quantum phase-flip code. In some embodiments, consistency with the stabilizer group of the base quantum code is the only constraint on a measurement code.

[0180]The general form of a measurement code can be specified by the parity check matrix H_M. This in turn is specified as a function of the (generally redundant) parity checks H_Z of the quantum repetition code and an additional set of s ancilla bits that label the measurements. Given H_Z, the parity check matrix of the measurement code is the block matrix H_M=(H_Z I_S), where I_S is the s×s identity matrix. Since there are s ancilla bits for readout, H_M is an s×(n+s) matrix. In some embodiments, the rows of the H_Z have even weight because the rows come from the stabilizers of a quantum repetition code. The rows are linearly independent, making the associated code have parameters [n+s, n, d] for some d≤n. The distance is never greater than n since a string of {circumflex over (Z)} operators on the data qubits, corresponding to 1's on exactly the first n bits, is always in the kernel of H_M.

[0181]The measurement of the j-th parity check in the measurement code can be done by a standard choice of circuit. In some embodiments, a CNOT gate is applied to the i-th qubit if there is a 1 in the i-th column, and target the ancilla labeled in column n+j. Note that by construction there is a 1 in position (j; n+j) of H_M. The effective error rate of this bare-ancilla measurement gadget depends on the number of CNOT gates used, and hence on the weight of the stabilizer being measured. Therefore, all other things (such as code distance) being equal, lower weight rows are preferred when designing a measurement code. The two examples considered here are generated from the following choices for H_Z, displayed here in transpose to save space:

$H_Z^T=\left(\begin{array}{ccc}1& 1& 0\\ 1& 0& 1\\ 0& 1& 1\end{array}\right),H_Z^T=\left(\begin{array}{ccccccccc}1& 0& 0& 0& 1& 1& 0& 0& 1\\ 1& 1& 0& 0& 0& 0& 1& 1& 0\\ 0& 1& 1& 0& 0& 1& 0& 1& 0\\ 0& 0& 1& 1& 0& 0& 0& 1& 1\\ 0& 0& 0& 1& 1& 0& 1& 1& 0\end{array}\right).$

[0182]These example codes saturate the distance bound, such that d=n for each code (e.g., d=3 and d=5, respectively). In contrast, the measurement code associated with repeating the measurements of the standard generators r times for n=r=3, is:

$H_Z^T=\left(\begin{array}{cccccc}1& 1& 1& 0& 0& 0\\ 1& 1& 1& 1& 1& 1\\ 0& 0& 0& 1& 1& 1\end{array}\right).$

[0183]Both this choice and the 3×3 choice above have distance d=3 as measurement codes. However, the 3×3 choice above corresponds to a [6, 3, 3] measurement code whereas the naive repeated generator method yields a [12, 3, 3] measurement code. In general, the naive scheme yields an [n_(n−1)r, n, d(n, r)] code, and for smaller r the distance will not yet saturate to n. For the case n=5 case, r must equal 2 before the measurement code has a distance 3, and r=4 before the distance saturates at d=5. Thus, the naive scheme yields either an [13, 5, 3] code or a [21, 5, 5] code, which are inferior in either distance or rate respectively to the [14, 5, 5] code that results from the choice. These examples also illustrate a counterintuitive feature of measurement codes, according to some embodiments. Consider again the naïve repeated generator method with n=5 and r=2 or 4. If the decoder works by first decoding the syndrome bits individually, then the data are only protected against at most (r−1)/2=0 or 1 arbitrary errors respectively. However, a decoder that uses the structure of the associated measurement code can correct 1 or 2 arbitrary data errors with these respective parameters, which then reduces the leading order behavior of the code failure probability.

[0184]Both of the two example codes above are small enough that the exact probability of a decoding failure can be computed via an exhaustive lookup table. To demonstrate the advantage of the measurement code over naive encoding and decoding, we estimate the probability of a logical error in the CNOT-gadget using the measurement code in the second example above for n=5. The corresponding threshold is ˜6×10⁻³. On the other hand, to reach a similar threshold using the naive decoder requires n=11, r=5. Thus, in some embodiments, the decoder requires fewer resources than the naive decoder. In general an optimal (maximum likelihood) decoder is infeasible to implement because it requires exponential resources in n and s to compute, so substantially larger codes will need decoding heuristics such as message passing algorithms to approach peak decoding performance. In some embodiments, the decoder declares failure whenever the data error is not guessed exactly right, even though this is not necessary. When repeated rounds of error correction occur, it is sufficient to define success as reducing the weight of any correctable error.

[0185]Methods of Performing QIP

[0186]Various methods of performing QIP are discussed above in connection with measuring error syndromes and performing bias-preserving gates. FIG. 16 is a flowchart of a method 1600 of performing QIP that applies generally to most of the embodiments described above that use a data qubit coupled to an ancilla qubit. In some embodiments, the physical realization of the data qubit and the ancilla qubit may be any of the physical systems described above.

[0187]At act 1602, the method 1600 includes driving an ancilla qubit with a stabilizing field. In some embodiments, the stabilizing field generates the asymmetry in the error channel of an ancilla qubit that is exploited to measure error syndromes and perform bias-preserving quantum gates. The stabilizing field may be applied to the ancilla qubit using the microwave field generator 160.

[0188]At act 1604, the method 1600 includes creating a Kerr-nonlinearity in the ancilla qubit using at least one Josephson junction of the ancilla. In some embodiments, coupling a superconducting circuit element to a cavity creates the Kerr-nonlinearity. For example, a transmon or a SNAIL may be located in a 3D cavity to create a Kerr-nonlinear cavity.

[0189]At act 1605, the method 1600 includes applying a plurality of microwave fields to the ancilla qubit and the data qubit. In some embodiments, these microwave fields may be applied to create pumped cat states in the Kerr-nonlinear cavity. In some embodiments, the microwave fields may be applied to perform rotation on the states of the data qubit or the ancilla qubit. In some embodiments, the microwave fields may be applied to perform conditional gates, such as conditional rotations, on one qubit based on the state of another qubit. In some embodiments, the microwave fields may be applied to couple the ancilla qubit to the data qubit or to couple the ancilla qubit to a readout cavity. Or, as discussed above, any number of operations may be performed by applying microwave fields to the data qubit and/or the ancilla qubit.

[0190]At act 1608, the method includes measuring the ancilla qubit. As discussed above, the ancilla qubit may be measured directly by, e.g., performing homodyne detection of a cavity of the ancilla qubit. Alternatively, the ancilla qubit may be measured by coupling the ancilla qubit to a readout cavity, conditionally displacing the state of the readout cavity based on the state of the ancilla qubit, and then measuring the state of the readout cavity. In some embodiments, the measurement of the ancilla is a QND measurement.

[0191]FIG. 17 is a flowchart of a method 1700 for performing readout of an ancilla qubit, according to some embodiments. In some embodiments, the method 1700 may be used to measure a property of a data qubit that is coupled to the ancilla qubit. In some embodiments, the method 1700 may implement a QND measurement.

[0193]At act 1704, the method 1700 includes turning off a stabilizing microwave field for an amount of time. In some embodiments, this allows the ancilla qubit to freely evolve. In some embodiments, the ancilla qubit may include a Kerr-nonlinear cavity and the state of the ancilla qubit may freely evolve under the Kerr-nonlinear Hamiltonian. In some embodiments, the free evolution of the ancilla qubit results in a rotation of the state of the ancilla qubit that could not be performed if the stabilizing field was still applied to the ancilla qubit.

[0194]At act 1706, the method 1700 includes re-applying the stabilizing microwave field to the ancilla qubit. In some embodiments, re-applying the stabilizing microwave field stops the free evolution of the state of the ancilla. In some embodiments, re-applying the stabilizing microwave field keeps the state of the ancilla in one of two coherent states. In some embodiments, re-applying the stabilizing field suppresses a particular type of error such that the error channel of the ancilla qubit is asymmetric. For example, the stabilizing field may suppress bit-flip errors.

[0195]At act 1708, the method 1700 includes applying an exchange microwave field to the ancilla qubit. In some embodiments, the exchange microwave field creates an interaction between the ancilla qubit and a readout cavity. In some embodiments, applying the exchange microwave field creates a three- or four-wave mixing interaction. In some embodiments, applying the exchange microwave field causes a Q-switch operation.

OTHER CONSIDERATIONS

[0196]Having thus described several aspects of at least one embodiment of this invention, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to be part of this disclosure, and are intended to be within the spirit and scope of the invention. Further, though advantages of the present invention are indicated, it should be appreciated that not every embodiment of the invention will include every described advantage. Some embodiments may not implement any features described as advantageous herein and in some instances. Accordingly, the foregoing description and drawings are by way of example only.

[0197]Various aspects of the present invention may be used alone, in combination, or in a variety of arrangements not specifically discussed in the embodiments described in the foregoing and is therefore not limited in its application to the details and arrangement of components set forth in the foregoing description or illustrated in the drawings. For example, aspects described in one embodiment may be combined in any manner with aspects described in other embodiments.

[0198]Use of ordinal terms such as “first,” “second,” “third,” etc., in the claims to modify a claim element does not by itself connote any priority, precedence, or order of one claim element over another or the temporal order in which acts of a method are performed, but are used merely as labels to distinguish one claim element having a certain name from another element having a same name (but for use of the ordinal term) to distinguish the claim elements.

[0199]All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms.

[0200]The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.”

[0201]As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified.

[0202]As used herein in the specification and in the claims, the phrase “equal” or “the same” in reference to two values (e.g., distances, widths, etc.) means that two values are the same within manufacturing tolerances. Thus, two values being equal, or the same, may mean that the two values are different from one another by ±5%.

[0203]The phrase “and/or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and/or” should be construed in the same fashion, i.e., “one or more” of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and/or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and/or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.

[0204]As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and/or” as defined above. For example, when separating items in a list, “or” or “and/or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of” or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e. “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of.” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law.

[0205]Also, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,” “comprising,” or “having,” “containing,” “involving,” and variations thereof herein, is meant to encompass the items listed thereafter and equivalents thereof as well as additional items.