A measurement-based fault-tolerant architecture for stubby code

Abstract

Systems and methods for performing fault-tolerant quantum operations for a four-legged cat code are provided. A quantum system includes an auxiliary qubit dispersively coupled to a first logical qubit, and the quantum system can be operated at least in part by generating and applying a first drive waveform to the auxiliary qubit, the first drive waveform including a first comb of pi pulses having selective frequencies corresponding to a first selection of even and odd cavity resonance frequencies of the first logical qubit; and reading out a state of the auxiliary qubit.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims the benefit under 35 U.S.C. §119(e) of U.S. Provisional Patent Application No. 63 / 293,034, filed on December 22, 2021, entitled “MEASUREMENT-BASED FAULT TOLERANT ARCHITECTURE FOR THE 4-LEGGED CAT CODE,” the entire contents of which are incorporated herein by reference.

[0003] Statement Regarding Federally Funded Research

[0004] This invention was made with Government support under Grant W911NF-18-1-0212 awarded by the U.S. Army Research Office. The Government has certain rights in this invention. Background Art

[0005] Quantum information processing techniques perform calculations by manipulating one or more quantum objects. These techniques are sometimes referred to as "quantum computing". In order to perform calculations, quantum information processors use quantum objects to reliably store and retrieve information. According to some quantum information processing methods, quantum simulations of classical computing "bits" (equal to 1 or 0) have been developed, which are called quantum bits or "qubits". A quantum bit can be composed of any quantum system with two different states (which can be considered as a 1 state and a 0 state), but it also has the special property that the system can be placed in a quantum superposition and therefore exist in these two states at the same time. Summary of the Invention

[0006] Some embodiments relate to a method of operating a circuit quantum electrodynamics system including an auxiliary qubit dispersively coupled to a first logical qubit. The method includes performing a quantum operation at least in part by generating and applying a first drive waveform to the auxiliary qubit, the first drive waveform including a first comb of pi pulses having selective frequencies corresponding to first selections of even and odd cavity resonance frequencies of the first logical qubit; and reading out a state of the auxiliary qubit.

[0007] Some embodiments relate to a quantum information processing system comprising: an auxiliary qubit; a first logical qubit dispersion-coupled to the auxiliary qubit; and at least one controller configured to perform a quantum operation at least in part by generating and applying a first drive waveform to the auxiliary qubit, the first drive waveform comprising a first comb of π pulses having selective frequencies corresponding to a first selection of even and odd cavity resonance frequencies of the first logical qubit; and reading out a state of the auxiliary qubit.

[0008] In some embodiments, the method includes generating and applying a second drive waveform to the auxiliary qubit before reading out the state of the auxiliary qubit, the second drive waveform comprising a second comb of π pulses having selective frequencies corresponding to a second selection of even and odd cavity resonance frequencies of the first logical qubit.

[0009] In some embodiments, the first selection includes selective frequencies 3χ, 4χ, 7χ, and 8χ, and the second selection includes selective frequencies 1χ, 2χ, 5χ, and 6χ.

[0010] In some embodiments, the circuit quantum electrodynamics system further includes a second logical qubit coupled to the first logical qubit by a first beam splitter, and the method further includes applying a third drive waveform to the first beam splitter to perform a detuned beam splitter interaction between the first logical qubit and the second logical qubit before reading out the state of the auxiliary qubit.

[0011] In some embodiments, performing a quantum operation includes generating a Bell state between a first logical qubit and a second logical qubit.

[0012] In some embodiments, performing a detuned beam splitter interaction between the first logical qubit and the second logical qubit includes performing a detuned beam splitter interaction between a first cavity resonator and a second cavity resonator.

[0013] In some embodiments, generating and applying the first drive waveform includes generating and applying a microwave waveform.

[0014] In some embodiments, generating and applying the first drive waveform includes generating the first drive waveform and applying the first drive waveform to a superconducting transmon.

[0015] In some embodiments, the method further includes generating the first four-qubit cluster state at least in part by applying a fourth drive waveform to a second beam splitter that couples the first logical qubit and the third logical qubit; and applying a fifth drive waveform to a third beam splitter that couples the second logical qubit to the fourth logical qubit.

[0016] In some embodiments, the method further includes generating the multi-qubit cluster state at least in part by applying a sixth drive waveform to a fourth beam splitter that couples the first logical qubit of the first four-qubit cluster state and the first logical qubit of the second four-qubit cluster state.

[0017] Some embodiments relate to a method of operating a circuit quantum electrodynamics system, the circuit quantum electrodynamics system including an auxiliary qubit dispersively coupled to a first logical qubit and a second logical qubit coupled to the first logical qubit by a first beam splitter. The method includes applying a first drive waveform to the auxiliary qubit, the first drive waveform including a π / 2 pulse; applying a second drive waveform to the first beam splitter to perform a detuned beam splitter interaction between the first logical qubit and the second logical qubit; applying a third drive waveform to the auxiliary qubit, the third drive waveform including a π / 2 pulse; and reading out a state of the auxiliary qubit.

[0018] In some embodiments, the circuit quantum electrodynamics system further includes a third logical qubit coupled to the first logical qubit by a second beam splitter, and the method further includes applying a fourth drive waveform to the second beam splitter to perform a detuned beam splitter interaction between the first logical qubit and the third logical qubit after applying the second drive waveform.

[0019] Some embodiments relate to a method of operating a circuit quantum electrodynamics system, the circuit quantum electrodynamics system including a first ancillary qubit dispersion-coupled to a first logical qubit and a second ancillary qubit dispersion-coupled to a second logical qubit, the first logical qubit being coupled to the second logical qubit by a first beam splitter. The method includes applying a first drive waveform to the first beam splitter to generate a resonant beam splitter interaction between the first logical qubit and the second logical qubit; and determining whether at least one of the first logical qubit and the second logical qubit is in a vacuum state by applying a second drive waveform to the first ancillary qubit to measure a state of the first logical qubit, and applying a third drive waveform to the second ancillary qubit to measure a state of the second logical qubit.

[0020] Some embodiments relate to a method of operating a circuit quantum electrodynamics system, the circuit quantum electrodynamics system including a first ancillary qubit dispersion-coupled to a first logical qubit, a second ancillary qubit dispersion-coupled to a second logical qubit, and a third logical qubit, the first logical qubit and the second logical qubit being coupled by a first beam splitter, and the second logical qubit and the third logical qubit being coupled by a second beam splitter. The method includes: preparing an arbitrary logical state in the first logical qubit; preparing a Bell state between the second logical qubit and the third logical qubit; and performing error correction on the arbitrary logical state by teleporting the arbitrary logical state from the first logical qubit to the third logical qubit, the teleporting including: introducing interference between the first logical qubit and the second logical qubit using the first beam splitter, and performing at least one measurement of the state of the first logical qubit and the second logical qubit using the first ancillary qubit after using the first beam splitter.

[0021] In some embodiments, preparing the Bell state includes: preparing a first coherent state in a second logical qubit; preparing a second coherent state in a third logical qubit; and performing a series of joint parity measurements on the second logical qubit and the third logical qubit.

[0022] Some embodiments relate to a circuit quantum electrodynamics system, comprising: an auxiliary qubit; and a plurality of logical qubits, the plurality of logical qubits comprising a first logical qubit dispersion-coupled to the auxiliary qubit, and a second logical qubit coupled to the first logical qubit by a beam splitter.

[0023] In some embodiments, the auxiliary qubit comprises a superconducting transporter qubit.

[0024] In some embodiments, the second logical qubit includes a plurality of logical qubits.

[0025] In some embodiments, a logical qubit in the plurality of logical qubits comprises a bosonic mode.

[0026] In some embodiments, the system further includes at least one controller configured to: prepare an arbitrary logical state in a first logical qubit; prepare a Bell state between a second logical qubit and a third logical qubit; and perform error correction on an arbitrary coherent state by teleporting the arbitrary logical state from the first logical qubit to the third logical qubit, the teleporting comprising: introducing interference between the logical qubit and the second logical qubit using at least one beam splitter; and performing at least one measurement of the state of the first logical qubit and the second logical qubit using the first auxiliary qubit and the second auxiliary qubit after using the at least one beam splitter. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Various aspects and embodiments are described with reference to the following drawings. The drawings are not necessarily drawn to scale. For clarity, not every component may be labeled in every drawing. In the drawings:

[0028] Figure 1 is a schematic diagram of an illustrative quantum information processing system according to some implementations of the technology described herein.

[0029] Figure 2 is a schematic diagram of another illustrative quantum information processing system in accordance with some embodiments of the technology described herein.

[0030] Figure 3A is a schematic diagram of an illustrative quantum circuit for fault-tolerant preparation of |+> states in a qubit, according to some embodiments of the technology described herein.

[0031] Figure 3B According to some embodiments of the technology described herein, Figure 3A Schematic diagram of an illustrative quantum information processing system of quantum circuits.

[0032] Figure 3C is a schematic diagram of an illustrative quantum circuit for performing parity measurements in accordance with some implementations of the technology described herein.

[0033] Figure 3D is a method for performing some embodiments of the technology described herein. Figure 3C Schematic diagram of illustrative drive waveforms for parity measurement.

[0034] Figure 4A is a schematic diagram of an illustrative quantum circuit for fault-tolerant preparation of |0> or |1> states in a qubit, according to some embodiments of the technology described herein.

[0035] Figure 4B is a schematic diagram of an illustrative quantum circuit for performing a Z measurement in accordance with some implementations of the technology described herein.

[0036] Figure 4C is a method for performing some embodiments of the technology described herein. Figure 4B Schematic diagram of illustrative drive waveforms for Z measurement.

[0037] Figure 5 is a schematic diagram of an illustrative quantum circuit for performing Z-basis fault-tolerant measurements in accordance with some implementations of the technology described herein.

[0038] Figure 6 is a schematic diagram of an illustrative quantum circuit for performing X-basis fault-tolerant measurements in accordance with some implementations of the technology described herein.

[0039] Figure 7 is a schematic diagram of an illustrative quantum circuit for performing XX-basis fault-tolerant measurements in accordance with some implementations of the technology described herein.

[0040] Figure 8A is a schematic diagram of an illustrative quantum circuit for performing ZZ-basis fault-tolerant measurements in accordance with some implementations of the technology described herein.

[0041] Figure 8B According to some embodiments of the technology described herein, Figure 8A Schematic diagram of an illustrative quantum information processing system of quantum circuits.

[0042] Figure 8C is a schematic diagram of an illustrative quantum circuit for performing a ZZ measurement in accordance with some implementations of the techniques described herein.

[0043] Figure 8D is a method for performing some embodiments of the technology described herein. Figure 8B Schematic diagram of an illustrative drive waveform for a ZZ measurement.

[0044] Figure 9A is a schematic diagram of an illustrative quantum circuit for performing ZZZ-basis fault-tolerant measurements in accordance with some implementations of the technology described herein.

[0045] Figure 9B According to some embodiments of the technology described herein, Figure 9A Schematic diagram of an illustrative quantum information processing system of quantum circuits.

[0046] Figure 9C is a schematic diagram of an illustrative quantum circuit for performing a ZZZ measurement in accordance with some implementations of the techniques described herein.

[0047] Figure 9D is a method for performing some embodiments of the technology described herein. Figure 9CSchematic diagram of an illustrative drive waveform for a ZZZ measurement.

[0048] Figure 10 is a flow diagram of a process 1000 for performing quantum operations in accordance with some implementations of the technology described herein.

[0049] Figure 11 is a schematic diagram of an illustrative quantum circuit for producing Bell states according to some embodiments of the technology described herein.

[0050] Figure 12 is a schematic diagram of an illustrative quantum circuit for performing telecorrection in accordance with some implementations of the technology described herein.

[0051] Figure 13 is a schematic diagram of an illustrative quantum circuit for preparing Greenberger-Horne-Zeilinger cluster states according to some embodiments of the technology described herein.

[0052] Figure 14 is a schematic diagram of another illustrative quantum circuit for preparing GHZ cluster states according to some embodiments of the techniques described herein.

[0053] Figure 15 is a schematic diagram of an illustrative quantum circuit for preparing |χ> states according to some embodiments of the technology described herein.

[0054] Figure 16 is a schematic diagram of an illustrative quantum circuit for teleporting a CNOT gate, according to some embodiments described herein.

[0055] Figure 17A is a method for preparing |Φ according to some embodiments of the technology described herein Had >Schematic diagram of a simplified quantum circuit for the state.

[0056] Figure 17B According to some embodiments of the technology described herein Figure 17A Detailed schematic of the quantum circuit.

[0057] Figure 18 is a schematic diagram of a quantum circuit configured to teleport a Hadamard gate according to some implementations of the technology described herein.

[0058] Figure 19 is a schematic diagram of an illustrative quantum circuit for a fault-tolerant implementation of a SWAP test between a first qubit and a second qubit in accordance with some embodiments of the techniques described herein.

[0059] Figure 20 is a schematic diagram of an illustrative quantum circuit configured to reduce errors present in a quantum state prepared with four qubits, according to some embodiments of the technology described herein.

[0060] Figure 21 is a schematic diagram illustrating the influence of the Kerr effect and χ′ on quantum states according to some embodiments of the technology described herein.

[0061] Figure 22A is a graph illustrating an example of a drive waveform generated using a frequency comb according to some implementations of the technology described herein.

[0062] Figure 22B is a diagram illustrating some embodiments of the technology described herein. Figure 22A Graph of the Fourier transform of the driving waveform.

[0063] Figure 23A is a graph illustrating another example of a drive waveform generated using a frequency comb according to some implementations of the technology described herein.

[0064] Figure 23B is a diagram illustrating some embodiments of the technology described herein. Figure 23A Graph of the Fourier transform of the driving waveform.

[0065] Figure 24 is a flow chart describing another process 2400 for performing quantum operations in accordance with some implementations of the technology described herein.

[0066] Figure 25A is a schematic diagram of another illustrative quantum circuit configured to produce Bell states with two qubits in accordance with some implementations of the technology described herein.

[0067] Figure 25B According to some embodiments of the technology described herein, Figure 25A Schematic diagram of a quantum circuit used to prepare a two-qubit ZZ Bell state cluster state.

[0068] Figure 26A is a schematic diagram of another illustrative quantum circuit configured to prepare a four-qubit cluster state in accordance with some embodiments of the techniques described herein.

[0069] Figure 26B According to some embodiments of the technology described herein, Figure 26A Schematic diagram of a four-qubit cluster state prepared using a quantum circuit.

[0070] Figure 27Ais a schematic diagram of another quantum circuit configured to generate a two-qubit entangled state according to some implementations of the technology described herein.

[0071] Figure 27B According to some embodiments of the technology described herein, Figure 27A Schematic diagram of the quantum circuit used to prepare the two-qubit entangled state.

[0072] Figure 28A is a schematic diagram describing another process for generating another four-qubit cluster state according to some embodiments of the techniques described herein.

[0073] Figure 28B is a schematic diagram illustrating the formation of XZZX cluster states according to some embodiments of the technology described herein.

[0074] Figure 29 is a schematic diagram of an illustrative conventional computer system in accordance with some implementations of the techniques described herein. DETAILED DESCRIPTION

[0075] Several different types of qubits have been successfully demonstrated in the laboratory. However, the lifetime of the state of many of these systems before information is lost due to decoherence of the quantum state or other quantum noise is currently around 100 μs. Despite this long lifetime, error correction techniques will be very important in quantum computing to enable reliable storage and retrieval of information stored in quantum systems. However, unlike traditional computing systems where bits can be copied for error correction purposes, cloning unknown states of a quantum system is not possible. However, the system can be entangled with other quantum systems that effectively spread the information in the system across several entangled objects.

[0076] The present application relates to improved quantum error correction techniques for correcting errors in the state of a quantum system that exhibits one or more bosonic modes. In this context, an "error" refers to a change in the state of a quantum system that may be caused by, for example, boson loss, boson gain, dephasing, time evolution of the system, etc., and an "error" changes the state of the system such that information stored in the system is altered.

[0077] As described above, quantum multilevel systems, such as qubits, exhibit quantum states that decoher within approximately 100 μs based on current experimental practice. Therefore, coupling the multilevel system with another system exhibiting a longer decoherence time can be beneficial. As will be described below, bosonic modes are particularly ideal for coupling to multilevel systems. Through this coupling, the state of the multilevel system can be represented by the bosonic mode instead, thereby retaining the same information in a longer-lasting state than would otherwise exist in the multilevel system alone.

[0078] The quantum information stored in the bosonic mode may still have a finite lifetime, so errors may still occur within the bosonic system. Therefore, when errors occur in its state, it would be desirable to manipulate the bosonic system to effectively correct these errors and thus regain the system's previous state. If a wide range of errors can be corrected, the state of the bosonic system can be preserved indefinitely (or at least for a long period of time) by correcting any type of error that may occur.

[0079] The fields of cavity quantum electrodynamics (cavity QED) and circuit QED represent one illustrative experimental approach to achieving quantum error correction. In these approaches, one or more qubit systems are each coupled to a resonator cavity in a manner that allows the quantum information contained in the qubit to be mapped to and / or from the resonator. The resonator typically has a longer stable lifetime than the qubit. The quantum state can later be retrieved from the qubit by mapping the state from the individual resonators back to the qubit.

[0080] When a multilevel system, such as a qubit, is mapped to the state of a coupled bosonic system, a specific method must be chosen to encode the qubit state in the bosonic system. This choice of encoding is often simply referred to as a "code."

[0081] As an example, the code can use the zero-boson state of the resonator to represent the ground state of the qubit, and use the one-boson state of the resonator to represent the excited state of the qubit. That is:

[0082]

[0083] Here, |g> is the ground state of the qubit, |e> is the excited state of the qubit, α and β are complex numbers representing the probability amplitude of the qubit being in state |g> or |e>, respectively, and |0> and |1> are the zero-boson and one-boson states of the resonator, respectively. While this is a perfectly efficient code, it is not robust to many errors, such as boson loss. That is, when boson loss occurs, it may be impossible to use this code to recover the state of the resonator before the loss.

[0084] The use of the code can be written more generally as:

[0085]

[0086] Among them, |W ↓ > and |W ↑ > is called a logical codeword (or simply "codeword"). The choice of code—equivalently, the choice of how to encode the state of a two-level system (e.g., a qubit) in the state of a bosonic system—thus involves the choice of |W ↓ > and |W↑ > to select a value.

[0087] When an error occurs, the state of the system changes to the resulting state, which is called the "error word" in this paper. and The superposition is as follows:

[0088]

[0089] Here, the index k refers to the specific error that has occurred. As mentioned above, examples of errors include boson loss, boson gain, dephasing, amplitude decay, and so on. In general, the choice of code affects the robustness of the system to errors. That is, when an error occurs, the code used determines the degree to which the previous state can be restored with fidelity. An ideal code will be able to associate with a wide range of errors, so that information is not lost when any error occurs, and any quantum superposition of logical codewords can be restored with fidelity.

[0090] However, one challenge faced by these approaches is that the code may be limited by the lifetime of the nonlinear assistance required for quantum control of the bosonic system. Typically, errors in the bosonic system are controlled and corrected by manipulating an auxiliary qubit coupled to the bosonic system. However, this may mean that when an error occurs in the auxiliary qubit, error correction of the state of the bosonic system may no longer be possible.

[0091] The inventors have recognized and appreciated that four-legged cat codes can provide a fault-tolerant platform for executing quantum computing operations in hardware-efficient quantum computing systems. In particular, the inventors have developed a universal set of operations for four-legged cat codes based on measurements of logical qubits and / or auxiliary qubits. This universal gate set maintains fault tolerance for the most likely first-order errors (including auxiliary decay and dephasing) in logical qubits and auxiliary qubits.

[0092] The inventors have developed a set of general operations based on fault-tolerant parity manipulation of bosonic systems. In particular, the inventors have extended the use of fault-tolerant parity measurements to enable non-destructive and fault-tolerant measurement of the Z, ZZ, and ZZZ logical operators in four-legged cat codes. The implementation of these logical operators involves measuring these operators using a detuned beamsplitter interaction while the helper is in a superposition state. In some embodiments, the ZZ and ZZZ operators can be measured even when the helper is directly coupled to only one of multiple logical qubits.

[0093] Using fault-tolerant parity measurements and the extensions discussed above, the inventors have also developed methods for preparing Z and X eigenstates, Bell states, and GHZ states in the four-legged cat code. In addition, the inventors have developed methods for performing robust measurements in Z, X, ZZ, and XX logical bases by combining beam splitter and cavity photon number measurements. For example, the implementation of the X measurement exploits the interference of coherent states on the logical state using beam splitter interactions. Thereafter, a photon number selective drive waveform is applied to the auxiliary qubit to determine whether one of the logical qubits (e.g., the cavity) is in a vacuum state. These measurements are fault-tolerant to all orders of superconducting transport decay and dephasing errors in the sense that the overall measurement error can be exponentially suppressed by repeating the measurement and taking a majority vote on the result.

[0094] The inventors also recognize and appreciate that, in combination with cavity shift operations, this set of operators is sufficient to perform Clifford operations in the four-legged cat code while maintaining first-order fault tolerance to quantum errors. To make this set universal, the inventors developed operations including fault-tolerant SNAP gates to realize arbitrary single-qubit Z rotations, or alternatively, operations including the preparation of high-fidelity arbitrary states on a single-qubit Bloch sphere through a distillation scheme. This will involve generating N incomplete copies of the target state and performing pairwise comparisons of the copies by performing non-destructive fault-tolerant SWAP tests between all possible pairs of copies. Post-selection is performed after all SWAP tests have passed, resulting in N copies of the state whose fidelity of the target state is higher than the fidelity with the initial state.

[0095] The inventors also recognized and appreciated that single-photon loss and transition-free backaction can be corrected in the four-legged cat code through a teleportation scheme ("stealth correction"). This scheme can be divided into two parts: creating suitable entangled Bell pairs; and performing measurements on the Bell basis. The inventors accordingly developed techniques for generating Bell states for the four-legged cat code and performing Bell measurements. Such Bell states are then used to correct for transition-free backaction, and the Bell measurements are teleported while simultaneously correcting for single-photon loss.

[0096] According to some embodiments, the codes described herein can be used to configure the state of a bosonic system. Bosonic systems can be particularly ideal systems in which to apply the techniques described herein because individual bosonic modes can exhibit equidistant spacing of coherent states. For example, a resonator cavity is a simple harmonic oscillator with equidistant horizontal spacing. Bosonic modes also contribute to quantum communication because they can be stationary for quantum memory, or contribute to interaction with conventional qubits, or they can propagate ("fly") for quantum communication (e.g., they can be captured and released from a resonator).

[0097] I. Illustrative Hardware Implementation

[0098] Figure 1 An illustrative system 100 suitable for practicing various aspects of the present application is depicted. In system 100, quantum system 101 includes an auxiliary qubit 110 coupled to a logical qubit 120 via dispersive coupling. That is, the detuning of the auxiliary qubit to the logical qubit is much greater (e.g., an order of magnitude greater) than the coupling strength between auxiliary qubit 110 and logical qubit 120. Logical qubit 120 is also coupled to logical qubit 140 by a beam splitter 130 (e.g., a programmable beam splitter). Energy source 150 may provide energy to one or both of auxiliary qubit 110, logical qubit 120, beam splitter 130, and / or logical qubit 140 to perform an operation on the system, such as preparing a state in any of logical qubits 120 and / or 140, measuring one or more of logical qubits 120 and / or 140, applying a gate operation to one or more of logical qubits 120 and / or 140, applying an operation to auxiliary qubit 110 or preparing a state in auxiliary qubit 110, detecting and / or correcting errors in auxiliary qubit 110 and / or logical qubits 120 and / or 140, or a combination thereof.

[0099] According to some embodiments, logical qubits 120 and logical qubits 140 can be implemented as any suitable multimode bosonic system. While this can include photonic systems, such as one or more microwave cavities, the techniques described herein are not limited to such systems. Logical qubits 120 and logical qubits 140 can be implemented as multimode bosonic systems, which can include any combination of multiple modes of a single bosonic system and / or single modes of multiple bosonic systems.

[0100] According to some embodiments, auxiliary qubit 110 may comprise any suitable quantum system having three different states, such as, but not limited to, a quantum system based on a superconducting Josephson junction, such as a charge qubit (Cooper-pair box), a flux qubit or a phase qubit, a superconducting transport qubit, or a combination thereof. Ancillary qubit 110 may be coupled to logical qubit 120 via dispersion coupling, which couples the state of auxiliary qubit 110 to the state of logical qubit 120. Logical qubit 120 may comprise any bosonic system that supports multiple bosonic modes, which may be implemented using any electromagnetic, mechanical, magnetic (e.g., quantized spin waves also known as magnons), and / or other techniques, such as, but not limited to, any cavity resonator (e.g., a microwave cavity). According to some embodiments, logical qubit 120 may comprise multiple transmission line resonators.

[0101] According to some embodiments, beam splitters 130 may be configured to provide a switchable beam splitter interaction between logical qubit 120 and one or more logical qubits 140. For example, each beam splitter 130 may be actuated between logical qubit 120 and one of logical qubits 140 in the form of The beam splitter 130 may be implemented using, for example, a superconducting microwave circuit including, but not limited to, four-wave mixing with a parametrically driven superconducting transporter and / or three-wave mixing with a superconducting nonlinear asymmetric inductor element-mon (“SNAILmon”) or a flux-pumped DC superconducting quantum interference device (“SQUID”).

[0102] System 100 also includes an energy source 150, a controller 160, and a storage medium 170 (e.g., a computer-readable storage medium). In some embodiments, a library 172 of pre-calculated drive waveforms can be stored on storage medium 170 and accessed by controller 160 to apply the waveforms to quantum system 101. For example, controller 160 can (e.g., in response to user input provided to the controller) access drive waveforms 172 stored on storage medium 170 and thereafter control energy source 150 to apply one or more drive waveforms to auxiliary qubit 110, logic qubit 120, beam splitter 130, and / or logic qubit 140, respectively.

[0103] As used herein, applying such an electromagnetic signal or pulse may also be referred to as "driving" the auxiliary qubit and / or the logic qubit. Coupling may utilize any technique to couple the auxiliary qubit and the logic qubit, for example by coupling the electric field and / or magnetic field generated by the auxiliary qubit and the logic qubit. According to some embodiments, the auxiliary qubit (e.g., a superconducting transporter) may be coupled to the logic qubit as a mechanical resonator via piezoelectric coupling. According to some embodiments, the auxiliary qubit (e.g., a superconducting transporter) may be coupled to a phonon, which in turn is coupled to a magnon via magnetostrictive coupling, to couple the auxiliary qubit to a logic qubit as a magnetic resonator.

[0104] Figure 2An alternative illustrative system suitable for practicing various aspects of the present application is depicted. In system 200, quantum system 201 includes an auxiliary qubit 110 coupled to a logical qubit 140 via dispersive coupling. Logical qubit 140 is also coupled to other logical qubits 140 by beam splitter 130. Beam splitter 130 is capable of switching on and off the beam splitter interaction between any pair of logical qubits 140. Energy source 150 can provide energy to one or both of auxiliary qubit 110, beam splitter 130, and / or logical qubit 140 to perform operations on the system, such as preparing a state in any one of logical qubits 140, measuring the state of one or more logical qubits 140, applying a gate operation to one or more logical qubits 140, applying an operation to auxiliary qubit 110, detecting and correcting errors in auxiliary qubit 110 and / or logical qubit 140, or a combination thereof.

[0105] II. Operations for the four-legged cat code

[0106] Bosonic quantum computing encodes quantum information in the degrees of freedom of simple harmonic oscillators. In this way, quantum error correction can be implemented in a hardware-efficient manner. That is, quantum errors occurring in the oscillator can be corrected without requiring much additional physical hardware. One such encoding is the four-legged cat code, which is designed to correct single-photon loss errors in oscillators, which are the main error channel in some quantum systems, such as quantum electrodynamics circuits.

[0107] To use this encoding as a quantum memory, it is necessary to prepare logical states in the appropriate codeword, detect and correct single-photon loss errors, and then read out the logical information from the quantum system. To further utilize this encoding for quantum computing, a universal gate set must also be implemented.

[0108] Without quantum control of the simple harmonic oscillator, neither quantum memory nor computation is possible. In order to achieve quantum control of the simple harmonic oscillator using classical external drive, a nonlinear source can be added to the system. For example, an auxiliary qubit (such as a superconducting transport qubit) can be added to the system, which is divergently coupled to the simple harmonic oscillator (e.g., a microwave cavity resonator). Unfortunately, the auxiliary qubit may be an additional source of errors that can propagate to the information stored in the simple harmonic oscillator.

[0109] Since these errors generated by the auxiliary qubit are of quantum nature, they can be described as transition operators. Although the possible quantum errors are endless, correcting the most likely errors that can occur in this cavity-superconducting transmission subsystem within the time window between error correction steps can significantly improve computing performance. Such errors include single photon losses in simple harmonic oscillators, single decays of excitations in auxiliary qubits, and / or dephasing of the state stored by the auxiliary qubit. This set of errors can be neatly summarized as:

[0110]

[0111] where |g> and |e> are the first two levels of auxiliary qubits, and is the annihilation operator of the simple harmonic oscillator. For the three-level auxiliary qubit, there is a similar set of errors:

[0112]

[0113] where |f> is the third level of auxiliary qubits.

[0114] If the sub-operations described in this article are designed so that when one of these errors occurs, it does not cause a logical error in the qubits in the simple harmonic oscillator, then the operation is fault-tolerant to these errors. This condition is met if either the error can be corrected at a later time or the error has a negligible effect on the logical information stored in the simple harmonic oscillator.

[0115] The inventors have recognized and appreciated that this level of fault tolerance required to achieve universal quantum computing using quadrilateral codes can be achieved in the measurement-based quantum computing (MBQC) paradigm. In circuit model quantum computing, gates are applied to qubits that remain fixed throughout the computation. In contrast, MBQC is performed by preparing qubits in entangled resource states, including multi-body entangled states, so-called "cluster states." Calculations can then be performed using cluster states by measuring qubits in a specific basis. Rather than directly implementing logic gates, quantum operations can be divided into the preparation and destructive measurement of quantum states; these operations are then used to implement quantum gates and quantum error correction.

[0116] The first quantum operation to achieve fault-tolerant quantum computation in the four-legged cat code is the simple harmonic oscillator (e.g., in conjunction with the Figure 1 or Figure 2 The state preparation in the logical quantum bit 120 or 140). Figure 3A is a schematic diagram of an illustrative quantum circuit 300 for fault-tolerant preparation of |+> states in a qubit, according to some embodiments of the technology described herein.

[0117] In some embodiments, quantum circuit 300 describes the operations applied to a single qubit in an order read from left to right. On the far left, the qubit starts in a vacuum state (|vac>). Thereafter, a shift 302 (D(α)) can be applied to shift the state of the qubit to a coherent state (e.g., α = 2 to 3). After the state of the qubit is shifted, a parity measurement 304 can be performed repeatedly. By requiring repeated parity measurements to obtain the same measurement results, tolerance to auxiliary errors is achieved. If each parity measurement 304 obtains a different result, it is inferred that an error has occurred and the state can be discarded. By requiring the two measurement results to be consistent, a state |α>±|-α> that is tolerant to a set of errors can be prepared.

[0118] In some embodiments, the Figure 3B An illustrative quantum information processing system 310 is shown that implements quantum circuit 300. Quantum information processing system 310 includes a logic qubit 312, depicted as a microwave cavity resonator. Logic qubit 312 is dispersion-coupled to an auxiliary superconducting transport qubit 314. A readout resonator 316 (e.g., a microwave strip resonator) is coupled to the auxiliary superconducting transport qubit 314 and is configured to provide input to and / or read information from the auxiliary superconducting transport qubit 314.

[0119] In some embodiments, as Figure 3C As depicted in the example of , parity measurement 304 can be described as a sequence of quantum operations applied to the auxiliary qubit and the logic qubit. Figure 3C In the quantum circuit of , the auxiliary qubit |g> is depicted under the logical qubit |ψ L >On the lower row. The quantum operation includes a first π / 2 rotation 304a of the auxiliary qubit, a unitary operation 304b applied to the logical qubit, a second -π / 2 rotation 304c of the auxiliary qubit, and a measurement 304d of the state of the auxiliary qubit.

[0120] like Figure 3D As shown in the example of , these quantum operations can be physically implemented by applying a series 320 of driving waveforms to the auxiliary quantum bit. The series 320 includes: a first sequence 322a, which includes a driving waveform including a g-eπ / 2 pulse and an e-fπ pulse; and a subsequent second sequence 322b, which includes a driving waveform including an e-fπ pulse and a g-eπ / 2 pulse. Sequences 322a and 322b are time delayed by T Π =π / χ spaced apart. After sequence 322b is completed, a readout 324 of the state of the auxiliary qubit is performed.

[0121] Figure 4A Another quantum operation for performing state preparation in the four-legged cat code is depicted in . Figure 4A In the example of FIG. 4 , according to some embodiments, the quantum circuit 400 is configured to prepare the four-legged cat state |a>±|iα>+|a>±|-iα>, which is used as the logical 0 and 1 codewords of the four-legged cat code. The quantum circuit 400 starts with the logical qubit in the vacuum state (|vac>). Thereafter, as combined with Figure 3A As described above, a displacement 302 (D(α)) can be applied. In order to prepare the cat-and-mouse state in a fault-tolerant manner, the following can be used: Figure 4A A series of parity measurements 304 and logical Z measurements 406 are applied in the order depicted in FIG.

[0122] In some embodiments, the logical Z measurement 406 can be implemented by measuring the 4-parity of the logical qubit. This measurement determines whether the logical qubit contains 0, 4, 8, etc. photons or 2, 6, 10, etc. photons. If the qubit contains 0, 4, 8, etc. photons, the measurement produces a result of +1; however, if the qubit contains 2, 6, 10, etc. photons, the measurement produces a result of -1. If the qubit contains an odd number of photons, the measurement produces a random result. Fault tolerance is again achieved by requiring that the parity measurement 304 and the logical Z measurement 406 pair agree for a successful state preparation attempt.

[0123] In some embodiments, logical Z measurement 406 can be described as a sequence of quantum operations applied to the auxiliary qubit and the logical qubit, such as Figure 4B The logical Z measurement 406 differs from the parity measurement 304 only in that the execution latency (T 4Π =π / 2χ=T Π / 2) is combined with Figure 3B half of the unitary operation 304b described above. Similarly, Figure 4C As depicted in FIG, the series of driving waveforms 410 is identical to the series of driving waveforms 320, with the only difference being the waiting time T between the first sequence 322a and the second sequence 322b. 4Π =π / 2χ.

[0124] In addition to preparing the states in logical qubits, the states of logical qubits must also be measured as part of quantum computing implementation. Figure 5 is a schematic diagram of an illustrative quantum circuit 500 for performing fault-tolerant measurements in a Z basis of a four-legged cat code according to some embodiments of the technology described herein. The quantum circuit 500 includes a logical qubit |ψ L> measurement 502. Such measurement 502 may be destructive because the measurement 502 dephases the state stored in the logical qubit when a decay error occurs during the measurement 502. In this case, although the state stored in the logical qubit cannot be used for further logical operations thereafter, the measurement can continue to improve the overall measurement fidelity by repeating the majority vote of the measurement results.

[0125] In some embodiments, measurement 502 in the Z basis of the quadruple code can be physically implemented by applying optimized control pulses to the auxiliary qubit to excite the auxiliary qubit only when and if the logical qubit contains n=0, 3, 4, 7, 8, ... photons. Alternatively, the auxiliary qubit can be driven with a linear combination of selective π pulses of appropriate frequencies to implement measurement 502.

[0126] Figure 6 is a schematic diagram of an illustrative quantum circuit 600 for performing fault-tolerant measurements in an X-basis of a quadruple-code, according to some embodiments. To perform fault-tolerant measurements in the X-basis, quantum circuit 600 includes the use of auxiliary logical qubits, as shown in the lower line of the circuit diagram. The auxiliary logical qubits can start in a vacuum state (|vac>) and can thereafter be shifted 602 (D(α)) (e.g., as combined with Figure 3C The displacement 302 is described in detail) and is prepared in a coherent state. By combining the coherent state with the logical quantum bit |ψ L >, and the measurement in the X basis distinguishes between the |α>±|-α> state and the |iα>±|-iα> state. Measurements 606 and 608 (e.g., using selective π pulses) determine whether only one of the logical qubit and the ancillary qubit contains a zero photon. If exactly one of the logical qubit and the ancillary qubit contains a zero photon, then we know that the input state is |α>±|-α> because when the input state is |iα>±|-iα>, only This is the intrinsic error probability, which can be very small if α is large enough. Like the measurement 502 in the Z basis, the measurement 600 in the X basis can also be fault-tolerant by repeating the measurement and using majority voting.

[0127] Figure 7 is a schematic diagram of an illustrative quantum circuit 700 for performing fault-tolerant measurements in the XX basis of a quadrilateral code, according to some embodiments. Figure 6 The measurement 600 is very similar. The measurement of quantum circuit 700 does not use a logical qubit and an auxiliary qubit, but rather uses two logical qubits |ψ L1 > and |ψL2 If one of the logical qubits is measured to contain a zero photon by measurements 606 and 608, this indicates that the two-legged cats are aligned in the same direction in phase space.

[0128] Figure 8A is a schematic diagram of an illustrative quantum circuit 800 for performing fault-tolerant measurements in a ZZ basis of a quadruple code, according to some embodiments. In some embodiments, quantum circuit 800 includes a measurement 802 of the joint 4-parity of two logical qubits.

[0129] One way to measure the joint 4-parity of two logical qubits is to couple the auxiliary qubit to a single cavity mode stored in the logical qubit. Then, a sequence of single mode 4-parity measurements can be performed without measuring any state. Then, as described in conjunction with the Figure 19 As described above, a SWAP operation is applied, followed by another single-mode 4 parity measurement. Afterwards, the auxiliary qubit can be measured in the X basis and another SWAP operation performed. However, this procedure faces the limitation that the beam splitter rate is typically less than χ, so that χ needs to be eliminated during the SWAP operation.

[0130] A faster sequence that avoids this problem is to combine the SWAP operation and the dispersion Hamiltonian into a single operation that implements a joint 4-parity measurement. To understand how this works, first note that the joint 4-parity operator is a joint rotation of the cavity phase space by 90 degrees: To measure this operator, a controlled joint cavity rotation can be applied between two π / 2 pulses, and then the auxiliary qubit is read out. The symmetric version of this gate can be written as:

[0131]

[0132] or equivalently:

[0133]

[0134] Note that each time this measurement is performed, there is an unconditional joint cavity phase rotation of π / 4 or 3π / 4, which can be tracked by software. With the correct timing and the ratio of χ to g, any unitary can be generated from the Hamiltonian:

[0135]

[0136] For a given value of χ, the two first operating points are set to or The above Hamiltonian can then be applied to the time

[0137] These specific ratios can achieve the desired unitary element. This measurement can be combined with Figures 3A to 3D The method for fault-tolerant parity measurement is similar to that described above, and is tolerant to superconducting transport errors. When measuring |e> using χ matching, the gf manifold using an auxiliary qubit allows detection of superconducting transport decay errors. This measurement does not dephase the cavity even in the presence of a single superconducting transport decay. Photon losses can be corrected if parity is tracked using a parity measurement and the 4-parity measurement is updated accordingly before the next parity transition. For example, if the cavity is odd, the auxiliary qubit can be read out in the y basis by adding a 90° phase shift to the final π / 2 pulse.

[0138] Return to Figure 8A , the total number of photons present in the two logical qubits is determined by combining the 4-parity measurement 802, for example, n=0, 4, 8, ... or n=2, 6, 10, .... Figures 3A to 3D As described, after measurement 802, parity measurements 304 are performed on each logical qubit. These parity measurements 304 are used to determine whether any of the logical qubits have experienced a photon loss.

[0139] In some embodiments, quantum circuit 800 may be used Figure 8B The illustrative quantum information processing system 810 shown in FIG. 8 is implemented as shown. The quantum information processing system 810 includes a first logical quantum bit 812a and a second logical quantum bit 812b. Both the first logical quantum bit 812a and the second logical quantum bit 812b can be microwave cavity resonators, such as Figure 8B As depicted in the example of FIG. 1 , a first logical qubit 812 a and a second logical qubit 812 b are coupled to each other by a beam splitter 814 . The first logical qubit 812 a is dispersion-coupled to an auxiliary superconducting transport sub-qubit 816 . A readout resonator 818 (e.g., a microwave strip resonator) is coupled to the auxiliary superconducting transport sub-qubit 816 and is configured to provide input to the auxiliary superconducting transport sub-qubit 906 and / or read out information from the auxiliary superconducting transport sub-qubit 816 .

[0140] In some embodiments, the measurement 802 of the joint 4-parity of two logical qubits can be described as a sequence of quantum operations applied to the auxiliary qubit and the two logical qubits, such as Figure 8C As depicted in the example of Figure 8C In a quantum circuit, the operation on the auxiliary qubit |g> is represented by the logical qubit |ψ L1 > and |ψ L2 >On the line below. Storage state |ψ L1The logical qubit of > is a logical qubit that is dispersion coupled to an auxiliary qubit, and the logical qubit |ψ L1 > and |ψ L2 >Coupled by beam splitters, such as Figure 8B The quantum operation includes: a first π / 2 rotation 802a of the auxiliary quantum bit, applied to the logical quantum bit |ψ L1 > and |ψ L2 >beam splitter operation 802b, a second -π / 2 rotation 802c of the auxiliary qubit, and measurement 802d of the state of the auxiliary qubit.

[0141] like Figure 8D As shown in the example, Figure 8C The quantum operation of can be physically realized by applying a series 820 of driving waveforms to the auxiliary qubit. The series 820 includes: a first sequence 822a, which includes a driving waveform including a g-eπ / 2 pulse and an e-fπ pulse; and a subsequent second sequence 822b, which includes a driving waveform including an e-fπ pulse and a g-eπ / 2 pulse. Sequences 822a and 822b are time-delayed. The time delay T ZZ During this time, a driving waveform is applied to the beam splitter coupling the two logical qubits to perform a detuned beam splitter interaction, which has a Hamiltonian of the following form:

[0142]

[0143] For time T ZZ ,in and g BS = +x / 2. In addition to measuring the joint 4-parity operator, this sequence also adds a deterministic rotation of -45° to the state stored in each logical qubit, which can be tracked by software. After sequence 822b is completed, a readout 826 of the state of the auxiliary qubit is performed.

[0144] Figure 9Ais a schematic diagram of an illustrative quantum circuit 900 for performing fault-tolerant measurements in the ZZZ basis of a four-legged cat code, according to some embodiments. Quantum circuit 900 can be viewed as an extension of quantum circuit 800 and includes 3-qubit measurements instead of 2-qubit measurements. To implement the ZZ measurement of quantum circuit 800, a switchable beam splitter interaction between a1 and a2 and a three-level auxiliary qubit dispersively coupled to a1 are used. To extend this to the ZZZ measurement of quantum circuit 900, an additional switchable beam splitter coupling can be added between logical qubit a1 and logical qubit a3, where a3 is the field operator of the third logical qubit. The target unitary element between π / 2 pulses is:

[0145]

[0146] Two paired beam splitter interactions implemented sequentially between a first logical qubit and a second logical qubit and then between the first logical qubit and a third logical qubit and then an appropriate wait time can be used to perform the above-mentioned unitary element.

[0147] Performing these two consecutive paired beam-splitter interactions yields the following unitary element:

[0148]

[0149] From this equation, we can see that the first logical qubit has accumulated an additional conditional phase. If a measurement is performed after performing these two beam splitter interactions, the operator Π1Z2Z3, where Π is the photon number parity of the first logical qubit, is obtained. To compensate for this, a waiting time T = π / (2χ) yields the following unitary element:

[0150]

[0151] After rearranging, we can get:

[0152]

[0153] Return to Figure 9A In some embodiments, quantum circuit 900 may use Figure 9B The illustrative quantum information processing system 910 shown in FIG. 910 includes a first logical quantum bit 912a, a second logical quantum bit 912b, and a third logical quantum bit 912c. Figure 9BAs depicted in the example of FIG, all three qubits 912a, 912b, and / or 912c can be microwave cavity resonators. The first logical qubit 912a and the second logical qubit 912b are coupled to each other by a beam splitter 914a. The first logical qubit 912a and the third logical qubit 912c are coupled to each other by another beam splitter 914b. The first logical qubit 912a is dispersion-coupled to an auxiliary superconducting transport sub-qubit 916. A readout resonator 918 (e.g., a microwave strip resonator) is coupled to the auxiliary superconducting transport sub-qubit 916 and is configured to provide input to the auxiliary superconducting transport sub-qubit 916 and / or read out information from the auxiliary superconducting transport sub-qubit 916.

[0154] In some embodiments, the measurement 902 of the three logical qubits can be described as a sequence of quantum operations applied to the auxiliary qubit and the three logical qubits, such as Figure 9C As depicted in the example of Figure 9C In a quantum circuit, the operation on the auxiliary qubit |g> is represented by the logical qubit |ψ L1 >、|ψ L2 > and |ψ L3 >On the following line. Store state |ψ L1 The logical qubit of > is a logical qubit that is dispersion-coupled to an auxiliary qubit, and the logical qubit pair |ψ L1 > and |ψ L2 > and |ψ L1 > and |ψ L3 >Each is coupled by a beam splitter, such as a combination of Figure 9B The quantum operation includes: a first π / 2 rotation 902a of the auxiliary quantum bit; an operation applied to the logical quantum bit |ψ L1 > and |ψ L2 >beam splitter operation 902b; applied to logical qubit |ψ L1 > and |ψ L3 >Second beam splitter operation 902c; applied to the first logical qubit |ψ L1 >unitary operation 902d; a second -π / 2 rotation 902e of the auxiliary qubit and a measurement 902f of the state of the auxiliary qubit.

[0155] like Figure 9D As shown in the example, Figure 9CThe quantum operation can be physically realized by applying a series 920 of driving waveforms to the auxiliary qubit and the beam splitter. The series 920 includes: a first sequence 922a, which includes a driving waveform including a g-eπ / 2 pulse and an e-fπ pulse; and a subsequent second sequence 922b, which includes a driving waveform including an e-fπ pulse and a g-eπ / 2 pulse. Sequences 922a and 922b are time delayed by 2T. ZZ +T 4Π Interval. Delay 2T at this time ZZ +T 4Π During this time, a driving waveform is applied to the two beam splitters that couple the logical qubits in pairs to perform a detuned beam splitter interaction with a Hamiltonian of the above form. 4Π The time period can be used to correct any rotation of the state stored in the logical qubits that have been accumulated (e.g., -90°, -45°, and -45° for the first logical qubit, the second logical qubit, and the third logical qubit, respectively).

[0156] Figures 9A to 9C ZZZ measurements complete the Clifford gate set of the four-legged cat code, as they can be used to implement CNOT gates when combined with the other quantum operations mentioned above. An entangled state |+++>+|---> can be created by performing ZZZ measurements on the separable state |+++>. Measuring pairs of ZZ operators (e.g., Z1Z2 and Z2Z3) on the same initial state |+++> creates similar entangled states |000>+|111>. Bell basis measurements on these states deterministically implement CNOT gates until local Pauli corrections occur, forming a set of gates known to be universal.

[0157] Figure 10 1 is a flow chart of a process 1000 for performing a quantum operation according to some embodiments described herein. Process 1000 can be used to operate a quantum information processing system, for example, including circuit quantum electrodynamics components. The quantum information processing system can include an auxiliary qubit (e.g., a superconducting transporter qubit, a SNAILmon qubit, an oscillator, or other qubit) coupled to a first logical qubit (e.g., a microwave cavity resonator). The first logical qubit can be coupled to a second logical qubit via a first beam splitter.

[0158] In some embodiments, process 1000 includes applying one or more drive waveforms to the auxiliary qubit and / or the first beam splitter. The drive waveforms can be stored on one or more computer-readable storage media (e.g., locally or remotely) and can be accessed by a controller. To apply the drive waveforms, the controller can cause an energy source (e.g., a microwave source) to generate the drive waveforms and transmit the drive waveforms to the auxiliary qubit and / or the first beam splitter.

[0159] In some embodiments, process 1000 can begin at act 1010, where a first drive waveform can be applied to an auxiliary qubit. The first drive waveform can include a π / 2 pulse. In some embodiments, the first drive waveform can include a sequence of drive waveforms. For example, the sequence of drive waveforms can include a g-eπ / 2 pulse and an e-fπ pulse.

[0160] In some embodiments, after act 1010, process 1000 can proceed to act 1020. In act 1020, a second drive waveform can be applied to the first beam splitter to perform a detuned beam splitter interaction between the first logical qubit and the second logical qubit. The detuned beam splitter interaction can be performed at During the time delay T ZZ During this time, a driving waveform can be applied to the beam splitter coupling the two logical qubits to perform a detuned beam splitter interaction, which has a Hamiltonian of the following form:

[0161]

[0162] For time T ZZ ,in and g BS =+χ / 2.

[0163] In some embodiments, after act 1020, process 1000 can proceed to act 1030, where a third drive waveform can be applied to the auxiliary qubit. The third drive waveform can include a π / 2 pulse. In some embodiments, the first drive waveform can include a sequence of drive waveforms. For example, the sequence of drive waveforms can include an e-fπ pulse and a g-eπ / 2 pulse.

[0164] In some embodiments, after act 1030, process 1000 may proceed to act 1040, where the state of the auxiliary qubit may be read out. In some embodiments, the state of the auxiliary qubit may be read out using a readout cavity or microwave strip resonator coupled to the auxiliary qubit. To read out the state of the auxiliary qubit, a measurement of the state of the auxiliary qubit may be performed. For example, a destructive measurement of the state of the auxiliary qubit may be performed. In some embodiments, this measurement may be performed using, for example, a microwave radiation detector that can distinguish between possible states of the readout cavity or microwave strip resonator. For example, in some embodiments, the microwave radiation detector may be a homodyne detector or a heterodyne detector.

[0165] III. Invisible Correction

[0166] In standard quantum teleportation, an unknown state is "teleported" to a new physical system. This teleportation can be achieved using two steps. First, an entangled Bell pair is created. Second, a measurement is performed on one half of the Bell pair and the unknown state in the Bell basis. Until a known Pauli correction (which depends on the measurement outcome) occurs, the unknown state is deterministically teleported to the other half of the Bell pair after this measurement.

[0167] One problem that has long been a prominent problem with four-legged cat codes is the so-called "transition-free" backaction, which causes the "size" α of the cat to decrease over time. If the correct Bell state can be created, this transition-free backaction can be mitigated by teleporting the quantum information to a new logical code with a larger α. For example, if the first logical qubit starts with a cat size of α0, and the first logical qubit has an energy loss rate κ c , then after time t, the cat will shrink to a valid

[0168] By generating a Bell state between a qubit with α=α′ in the logical basis and a qubit with α=α0 in the logical basis, a suitable Bell state can be created to correct for the transition-less backaction. Figure 11 Quantum circuit 1100 illustrates the fault-tolerant preparation of Bell states for this purpose according to some embodiments.

[0169] Quantum circuit 1100 begins by preparing two arbitrary states in two logical qubits. A first logical qubit may be shifted by a displacement 1102 (D1(α)) and a second logical qubit may be shifted by a displacement 1104 (D2(β)), thereby preparing two quantum states in the first logical qubit and the second logical qubit. In some embodiments, the first logical qubit and the second logical qubit may initialize their states in different logical bases. Figure 11In the example of [ 1 ], the first logical qubit is in a cat code of "size" α, while the second logical qubit is in a cat code of size β. Preparing the two logical qubits in different logical bases enables correcting for transition-free backaction.

[0170] Afterwards, if combined Figures 3A to 3D As described above, two parity measurements 304 may be performed, one for each of the first logical qubit and the second logical qubit. The quantum circuit 1100 may then proceed to perform two consecutive ZZ measurements, such as in combination with Figures 8A to 9C Thereafter, two additional parity measurements 304 may be performed on each of the first logical qubit and the second logical qubit. As described herein, to ensure fault-tolerant generation of Bell states, the first parity measurement 304 and the second parity measurement 304 must be consistent for each of the first logical qubit and the second logical qubit. Furthermore, the two ZZ measurements 802 must also be consistent to ensure fault tolerance.

[0171] The prepared Bell state 1100 can then be used to measure the logical qubit |ψ by performing measurements in the Bell basis. L >_α performs invisible correction, such as Figure 12 First, the first qubit in Bell state 1100 (prepared with a cat code of size α) can be connected to the logical qubit |ψ L >_α. Then, both the first qubit of Bell state 1100 and the logical qubit can be measured in the Z basis of the four-legged cat code using measurement 1204. In some embodiments, measurement 1204 can be equivalent to the method described herein in conjunction with Figure 5 The measurement 502 is described. Thereafter, both the first qubit of Bell state 1100 and the logical qubit can be measured in the XX basis of the four-legged cat code using measurement 1206, which can be equivalent to the measurement described herein in conjunction with Figure 6 The measurements 606. These measurements would then be stored in the logical qubit |ψ L > is teleported to a second qubit in Bell state 1100 (with a cat code of size β), thereby correcting the transition-free backaction and preventing leakage errors from accumulating over multiple quantum operations.

[0172] Since the beam splitter preserves the total photon number parity (i.e., the photon number is preserved), the ZZ information can still be extracted by measuring the local photon number parity mod (4) and summing the results to determine the ZZ information. The protocol is fault-tolerant because after the beam splitter, all logical XX and ZZ information has been mapped into the cavity's non-local photon number space. Although an auxiliary qubit error may still dephase the logical qubit during the stealth correction process, at least two photons need to be lost in either cavity to produce an incorrect measurement result.

[0173] In particular, for cluster state models of quantum computing, a potentially useful subroutine is the creation of Greenberger-Horne-Zeilinger (GHZ) entangled states such as |000>+|111> and |+++>+|--->. Figure 13 1 is a schematic diagram of an illustrative quantum circuit 1300 for preparing a |000>+|111> GHZ cluster state, according to some embodiments described herein. Quantum circuit 1300 begins by preparing three arbitrary states in three logical qubits by applying shifts 302 (D1(α), D2(α), D3(α)) to each logical qubit. Thereafter, a parity measurement 304 is performed on each logical qubit. A first pair of ZZ measurements 802 is performed on the first qubit and the second qubit, and then a second pair of ZZ measurements 802 is performed on the second qubit and the third qubit. Finally, a parity measurement 304 is performed on each logical qubit. As previously described, to provide fault tolerance, the first and final parity measurements must agree, the first pair of ZZ measurements 802 must agree, and the second pair of ZZ measurements 802 must agree to prepare the |000>+|111> GHZ cluster state.

[0174] Figure 14 is a schematic diagram of another illustrative quantum circuit 1400 for preparing a |+++>+|--->GHZ cluster state according to some embodiments described herein. Quantum circuit 1400 is similar to Figure 13 The quantum circuit 1300 is the same as that of FIG1 , but instead of two pairs of ZZ measurements 802, a single pair of ZZZ measurements 902 is performed between the two sets of parity measurements 304. In this case, each ZZZ measurement 902 must be consistent to maintain fault tolerance.

[0175] Combining GHZ state preparation with Bell state measurement enables the execution of fault-tolerant CNOT gates between logical qubits. Like most teleportation gate protocols, this can be broken down into two steps: creating a suitable entangled state, then performing a Bell measurement between that entangled state and the logical qubit to simultaneously teleport information to the remaining unmeasured qubit, and then executing the gate. This gate can be executed until some Pauli correction occurs (depending on the measurement outcome).

[0176] To prepare a CNOT gate, we first prepare the |χ> state, such as Figure 15 The example of quantum circuit 1500 is depicted. The |χ> state can be described as where |Ψ> is a Bell state To prepare the |χ> state, two different types of GHZ states are fused together using a fault-tolerant Bell measurement 1200. A Bell measurement between qubits 3 and 4 projects the remaining qubits into the |χ> state, until a local Pauli operation is determined by the Bell measurement result. This is equivalent to building a larger cluster state from two smaller building blocks.

[0177] According to some embodiments, once the |χ> state is prepared, it can be used to teleport a CNOT gate, such as Figure 16 A quantum circuit 1600 is shown. CNOT gates can be teleported using Bell measurements 1200 between pairs of logical qubits 1 and 2, and logical qubits 5 and 6. The output of quantum circuit 1600 is stored as logical information on the unmeasured qubits 3 and 4, and the Bell measurement results indicate which (if any) Pauli corrections should be applied to the output CNOT|ψ2ψ1>. It will be appreciated that there may be more efficient ways to compile quantum circuits with CNOT gates, reducing the number of operations and measurements, but this explicit construction is useful for demonstrating that the set of operations described herein is indeed universal.

[0178] Combining state preparation with Bell state measurement also enables the performance of a fault-tolerant Hadamard gate between two logical qubits. Figure 17A is a method for preparing Hadamard states |Φ according to some embodiments Had >Schematic diagram of quantum circuit 1700. Figure 17B An expanded version of this quantum circuit 1700 is depicted in .

[0179] The precursor two-qubit entangled state is an eigenstate of the XZ operator, represented by |Φ Had >. This state can also be written as |0+>±|1->, |+i+i>±i|-ii>, or H2|ψ 12 >. Quantum circuit 1700 utilizes three logical qubits that are initially in the |vac\ket state. Each qubit is shifted into a coherent state by a displacement 302, and a rotation 1706 of π / 2 places the three logical qubits in the |+i> state. In some embodiments, the rotation 1706 can be performed using a fault-tolerant SNAP gate or a switchable Kerr gate. A ZZZ measurement 902 is first performed on the three logical qubits using a fault-tolerant parity measurement 304 and a ZZZ measurement 902, resulting in the state |+i+i+i>±|-iii>. Inconsistency in the set of measurements indicates that a first-order error has occurred and the protocol should be restarted.

[0180] Thereafter, one of the qubits is destructively measured in the X basis using measurement 600, which utilizes an additional ancillary qubit initialized in a different logical basis than the three logical qubits. Measurement 600 includes performing a beam splitter interaction 1708 between one of the logical qubits and the ancillary qubit, and then destructively measuring the states of the logical qubit and the ancillary qubit using measurement 606. Performing a destructive measurement on the logical qubit in the X basis projects the two-qubit state of the other two logical qubits onto the state |+i+i>±i|-ii>, where the sign is determined by both the results of ZZZ measurement 902 and X measurement 600.

[0181] In preparation |Φ Had > state, it can be used to teleport a single-qubit Hadamard gate to another logical qubit, such as Figure 18 In some embodiments, the quantum circuit 1800 includes a quantum circuit having |ψ L >Logical qubits and double qubits of the state |Φ Had By performing fault-tolerant Bell measurement 1200 between qubits in the state |Φ Had >state. After performing 1200 fault-tolerant Bell measurements, the two-qubit |Φ Had The second qubit of the state can now store the quantum state

[0182] Combine Figures 17A to 18 The described protocol utilizes a minimum of five logical qubits (e.g., five microwave cavity resonators), with an auxiliary qubit coupled to each logical qubit. It should be recognized that this implementation of the Hadamard gate is not particularly hardware efficient, but combined with CNOT and R Z θ operation, it can be seen that the above quantum operation set is universal.

[0183] IV. State Purification Using the SWAP Test

[0184] Purification via SWAP testing refers to a general method for symmetrizing general qubits. The inventors have recognized and appreciated that this method can be used to prepare states in bosonic qubits with high fidelity. In particular, when using error-prone (e.g., noisy) procedures to generate several copies of a target state, non-destructive SWAP testing between pairs of states can be used to reduce errors. The SWAP test results can then be post-selected to reduce errors in state generation.

[0185] This is an independent procedure that can be used for general state preparation in bosonic modes to reduce the impact of random errors in state preparation. The preparation of |±i> and |T> states in the measurement-based scheme described here may be particularly useful as an alternative to using fault-tolerant SNAP gates as non-Clifford operations. Instead of implementing direct fault-tolerant gates (such as SNAP gates), fault-tolerant measurements can be used to purify noisy states generated by other means (such as optimal control pulses or state transfer from auxiliary qubits to logical qubits). The advantage of this approach is that the noise channel can be complex and different for each input cavity state.

[0186] The SWAP test measurement is tolerant to first-order errors, and therefore the initial state preparation error can be much larger than the SWAP test error. Under such conditions, the SWAP test can be used to purify the initial state and reduce the state preparation error. The process begins by preparing N replicas of the desired quantum state to be initialized. For simplicity, it can be assumed that the probability of some error in the state preparation is p err , and the probability of no error is (1-p err ). When a SWAP test measurement is performed between two of these cavities, the measurement result indicates the probability of failure p err / 2 is small, so the protocol must be restarted. But most of the time, the SWAP test measurement will succeed, producing two logical qubits with an error probability of p err / 2. This direct trade-off between probability of success and state fidelity is very favorable.

[0187] By repeating the SWAP test for all different pairings of cavities, when the SWAP test measurement succeeds, the error probability can be reduced until it reaches a limit set by the fidelity of the SWAP test measurement. Because the SWAP test can be performed with fault tolerance, this technique can, in principle, be used to prepare cavity states with high fidelity.

[0188] To illustrate this approach, the construction of a single fault-tolerant SWAP test measurement from a common operation is described. Figure 19 is a schematic diagram of an illustrative quantum circuit 1900 for implementing a fault-tolerant SWAP test between a first qubit and a second qubit, according to some embodiments. A SWAP test in this context is a non-destructive measurement of the SWAP operator between two logical qubits. If |ψ1> and |ψ2> are initial input states, then the SWAP test will project these states onto , because the symmetric and asymmetric superpositions are ±1 eigenstates of the SWAP operator.

[0189] To perform this measurement, a 50-50 beam splitter interaction 1900a is first performed between the two logical qubits. Afterwards, the parity of the number of photons in one of the patterns is measured using a parity measurement 304 within the “beam splitter” framework. Typically, a parity measurement on a single logical qubit measures the parity operator But in the beam splitter frame, Transformation so that the parity measurement 304 measures the SWAP operator After performing the parity measurement 304, another 50-50 beam splitter interaction 1900b is performed. The final beam splitter 1900b is an inverted 50-50 beam splitter implemented by inverting the phase of one of the beam splitter pumps. This sequence of operations is equivalent to a parity measurement in the beam splitter framework.

[0190] Taken at face value, the results of the SWAP test are straightforward to interpret. If the result obtained is +1 (i.e., the ancillary qubit is in the |g> state), then the two input states are more likely to be the same |ψ1> and |ψ2> and therefore error-free. By post-selecting on this result, the probability of any state being an error is correspondingly reduced.

[0191] The probability of obtaining a result of ±1 is 1±|<ψ1|ψ2>| 2 / 2. If one of the states experiences an error during initial preparation, then among the various possible errors, it is possible that |<ψ1|ψ2>| = 0. Furthermore, obtaining a result of +1 does not guarantee that an error has not occurred. If |<ψ1|ψ2>| = 0, there is still a probability of 0.5 that a result of +1 will be obtained. Therefore, when the SWAP test passes, the errors in the two cavity states are halved, but never completely eliminated.

[0192] The density matrix form can express this more precisely. The initial noisy cavity state can be written as:

[0193] ρ init =(1-p err )|ψ t ><ψ t |+p err ∑ i p i |ψ i ><ψ i |where, |ψ t > is the target state to be prepared with high fidelity, and |ψ i > is the state obtained when an error occurs in the initial preparation, where <ψ i |ψ t >=0, and p i is a real scalar that sums to 1.

[0194] The initial dual-cavity state can be written as Obtaining a +1 result is equivalent to applying the projection operator (1+SWAP) / 2. If the partial trace is in p err Take the first order, then the state of each logical quantum bit will be ρ final ,in:

[0195] ρ final =(1-p err / 2)|ψ t )<ψ t |+p err / 2∑ i p i |ψ i ><ψ i |

[0196] Figure 20 is a schematic diagram of an illustrative quantum circuit 2000 configured to reduce errors in quantum states prepared in four qubits according to some embodiments of the technology described herein. The quantum circuit 2000 includes several SWAP tests 1900 and SWAP operations 2002 between pairs of cavities. Figure 20 In the example, the process starts with ρ init During the implementation of quantum circuit 2000, SWAP tests 1900 can be performed between all six permutations of pairs of logical qubits to reduce the error of each state to p err / 8. Using ρ init Additional copies can further reduce the error rate at the cost of adding more SWAP tests and SWAP operations. Figures 8A to 8D As described, this protocol can be experimentally implemented with the same hardware as for ZZ measurements.

[0197] V. Correction for the Kerr Effect and χ'

[0198] In some quantum information processing schemes, it is desirable to take into account additional effects that may introduce perturbations into the quantum system. For example, the Kerr effect and the χ' effect may cause perturbations to the ZZ and / or ZZZ measurements described herein, making them less robust. These effects are particularly pronounced for systems that utilize a large number of photons (e.g., greater than or equal to 10 photons), because the frequency of transitions between the states of the measured auxiliary qubits depends on the number of photons stored in the logical qubit. For example, the χ' effect scales quadratically with the number of photons stored in the logical qubit, making the χ' effect more difficult to distinguish and correct for the larger the number of photons. Consideration of these effects is particularly important in MBQC, where a larger number of photons are used to perform the computational process.

[0199] The Kerr effect and the influence of χ' can be described by the last two terms of the following two-qubit Hamiltonian:

[0200]

[0201] Figure 21 Schematic diagram of a Bloch sphere illustrating the effects of the Kerr effect and χ' on quantum states, according to some embodiments. These two effects result in perturbations 2102 and 2104 around the Bloch sphere, which may reduce the robustness of the ZZ and / or ZZZ measurements described herein and increase the probability of decoherence in quantum circuits.

[0202] To counteract these effects, an alternative process can be used to prepare the quantum state stored in the logical qubit and similarly alter the quantum operation. In some embodiments, the cat state can be prepared by first shifting the state of the logical qubit from the vacuum state |vac> to the state |α>. Thereafter, the |α> state can be driven to |0> using a drive waveform comprising a selective g-fπ pulse comb. L A selective g-fπ pulse comb can include π pulses at multiple frequencies corresponding to frequencies (0x, 4x, 8x, 12x, ...). Using these selective frequencies in a g-fπ pulse comb can address the effects of χ', while varying the phases of the component π pulses can address Kerr effect perturbations because these phases provide a quadratic correction to the equal spacing of the energy levels of the logical qubits. Figure 22A An example of a selective g-fπ pulse comb is shown in, Figure 22B The corresponding Fourier spectrum is shown in .

[0203] In some embodiments, the measurement can also be adjusted to offset the effects of the χ' and Kerr effects. For example, XX and ZZ information can be extracted simultaneously to perform a Bell measurement using a three-level auxiliary qubit (such as a three-level superconducting transporter qubit). Three measurements can be performed to extract this information. In some embodiments, these measurements can be performed simultaneously. First, selective Raman conversion can be used to measure information associated with the |f> state. Second, information associated with the |e> state can be measured by driving the auxiliary qubit with a drive waveform including π pulses, which π pulses include a selective frequency comb with frequencies (3χ, 4χ, 7χ, 8χ, ...). Third, information associated with the |g> state can be measured by using a drive waveform including π pulses, which π pulses include a selective frequency comb with frequencies (1χ, 2χ, 5χ, 6χ, ...). Figure 23A An example of such a driving waveform comprising two frequency combs is shown in to demonstrate the |e> state and the |g> state. Figure 23B The corresponding Fourier transform is shown in .

[0204] Figure 24 24 is a flow chart describing another process 2400 for performing quantum operations according to some embodiments of the technology described herein. Process 2400 can be used to operate a quantum information processing system, for example, including circuit quantum electrodynamics components. The quantum information processing system can include an auxiliary qubit (e.g., a superconducting transporter qubit, a SNAILmon qubit, an oscillator, or other qubit) coupled to a first logical qubit (e.g., a microwave cavity resonator).

[0205] In some embodiments, process 2400 can begin at act 2410, where a first drive waveform is generated and applied to an auxiliary qubit. The drive waveform described in conjunction with process 2400 can be stored on one or more computer-readable storage media (e.g., locally or remotely) and can be accessed by a controller. To apply the drive waveform, the controller can cause an energy source (e.g., a microwave source) to generate the drive waveform and transmit the drive waveform to the auxiliary qubit and / or other components of the quantum information processing system.

[0206] In some embodiments, the first drive waveform includes a first comb of π pulses having selective frequencies corresponding to a first selection of even and odd cavity resonance frequencies of the first logical qubit. For example, the first comb of π pulses can have selective frequencies corresponding to (3x, 4x, 7x, 8x, ...) frequencies.

[0207] In some embodiments, the method optionally includes performing operation 2420 before reading out the state of the auxiliary qubit. Action 2420 may include generating a second drive waveform and applying the second drive waveform to the auxiliary qubit. The second drive waveform may include a second comb of π pulses having a selective frequency corresponding to a second selection of even and odd cavity resonance frequencies of the first logical qubit. In some embodiments, the second comb of π pulses may have a selective frequency corresponding to a selective frequency of (1x, 2x, 5x, 6x, ...).

[0208] In some embodiments, after act 2410 or 2420, process 2400 may proceed to act 2440, where the state of the auxiliary qubit may be read out. In some embodiments, the state of the auxiliary qubit may be read out using a readout cavity or microwave strip resonator coupled to the auxiliary qubit. To read out the state of the auxiliary qubit, a measurement of the state of the auxiliary qubit may be performed. For example, a destructive measurement of the state of the auxiliary qubit may be performed. In some embodiments, such a measurement may be performed using, for example, a microwave radiation detector capable of distinguishing between possible states of the readout cavity or microwave strip resonator. For example, in some embodiments, the microwave radiation detector may be a homodyne detector or a heterodyne detector.

[0209] In some embodiments, performing a quantum operation includes measuring a Bell state between a first logical qubit and a second logical qubit. In such embodiments, the quantum electrodynamics system further includes a second logical qubit coupled to the first logical qubit via a first beam splitter. For example, the first logical qubit and the second logical qubit may each be a microwave cavity resonator coupled by the first beam splitter. The method may include applying a third drive waveform to the first beam splitter to induce a detuned beam splitter interaction between the first logical qubit and the second logical qubit before reading out the state of the auxiliary qubit. Thereafter, process 2400 may proceed to operation 2440 described above.

[0210] In some embodiments, process 2400 additionally includes generating a first four-qubit cluster state. The four-qubit cluster state can be generated at least in part by applying a fourth drive waveform to a second beam splitter coupling the first logical qubit and the third logical qubit to induce a beam splitter interaction between the first logical qubit and the third logical qubit. Furthermore, the four-qubit cluster state can be generated by applying a fifth drive waveform to a third beam splitter coupling the second logical qubit to the fourth logical qubit. In this manner, the quantum states stored in the four logical qubits can be entangled to create the four-qubit cluster state.

[0211] In some embodiments, process 2400 additionally includes generating a multi-qubit cluster state. For example, the multi-qubit cluster state can be an XZZX cluster state as described herein, or can be any other multi-qubit cluster state suitable for MBQC. The multi-qubit cluster state can be generated at least in part by applying a sixth drive waveform to a fourth beam splitter coupling a first logical qubit of the first four-qubit cluster state and a first logical qubit of the second four-qubit cluster state.

[0212] VI. Cluster Preparation

[0213] The inventors have recognized and appreciated that the above quantum operations can be used to generate cluster states suitable for MBQC. Once the cluster state is generated, computations can be performed by measuring qubits in certain bases. Alternatively or additionally, the cluster state is useful for quantum communication and networking.

[0214] Figure 25A is a schematic diagram of an illustrative quantum circuit 2500 configured to produce Bell states in two qubits, according to some embodiments. Figure 25B 25 is a schematic diagram of a two-qubit ZZ Bell state 2510 , which may be prepared using a quantum circuit 2500 in some embodiments. Figure 25BThe diagram of includes two qubits 2512 fabricated in a first logical base, represented by closed circles. The line connecting the two qubits 2512 represents coupling through entanglement.

[0215] The quantum circuit 2500 begins with two logical qubits prepared in the |α> state and the |iα> state, respectively. The two logical qubits are coupled by a beam splitter, and the quantum circuit 2500 includes creating a beam splitter interaction 2504 between the two logical qubits. Before and after the beam splitter interaction 2504, parity measurement 304 is used to ensure fault tolerance. If Π1+Π2=Π3+Π4, then the Bell state |Φ Bell >The creation was successful.

[0216] like Figure 26A As shown, an example of a four-qubit cluster state can be created by chained beam-splitter interactions. Figure 26B is a schematic diagram of a four-qubit cluster state 2610 that can be generated using quantum circuit 2600. Figure 26A The quantum circuit 2600 first starts at and In the example, two logical qubits prepared in the |0> state are first measured 304 for each of the two logical qubits, and then a beam splitter interaction 2604a is performed between the two logical qubits. Thereafter, each of the two initial logical qubits is coupled to two additional logical qubits prepared in the |0> state via beam splitter interactions 2604b and 2604c. Thereafter, a second parity measurement 304 is performed on all four logical qubits. If π1+π2=π3+π4+π5+π6, then the generation of the four-qubit cluster state is successful.

[0217] Another building block of MBQC's cluster state is a two-qubit cluster consisting of two qubits, each of which is prepared by teleporting a Hadamard state in a different logical basis. Figure 27A is a schematic diagram of a quantum circuit 2700 configured to generate Figure 27B Depicted two-qubit entangled state 2710. Two-qubit entangled state 2710 includes a first qubit 2512 prepared in a first logical basis (eg, X) and a second qubit 2714 prepared in a second, different logical basis (eg, Z).

[0218] In some embodiments, quantum circuit 2700 is used in Two logical qubits are prepared in the |Φ state and the |0> state, and a beam splitter interaction is first performed between them 2702. Thereafter, a parity measurement 304 is performed on each logical qubit, resulting in |++>+i|-->. Then, a fault-tolerant SNAP operation is applied once to each logical qubit 2704, resulting in |Φ Had >=|+i+i>+i|-ii>≡|0+>+|1->state.

[0219] Figure 28A is a schematic diagram illustrating a fusion process according to some embodiments of the techniques described herein that can be used to generate another four-qubit cluster state. The process can begin at stage 2800 with four separate cluster states, including three two-qubit states and one four-qubit state. Bell measurements 2802 (which can be any suitable Bell measurement described herein) can be used to "fuse" the qubits of each of these smaller resource states to produce a four-qubit cluster state 2810.

[0220] Quantum operations (such as combining Figures 25A to 28B The quantum operations described herein can be further chained to generate larger cluster states useful for MBQC. For example, such cluster states can include the XZZX cluster states described herein, or, alternatively or additionally, RHG cluster states. Figure 28B is a schematic diagram illustrating the formation of XZZX cluster states according to some embodiments of the technology described herein.

[0221] like Figure 28B As shown in the example of , four-qubit cluster states 2610 and 2810 can be fused to form larger cluster states, such as cluster state 2820. These larger cluster states can be further fused to create final cluster states for MBQC or other applications. Figure 28B As depicted in FIG, in some embodiments, the larger cluster state can be an XZZX cluster state 2830. Additional aspects of the XZZX cluster state are described in J. Claes, J. Eli Bourassa, and S. Puri, “Tailored cluster states with high threshold under biased noise,” submitted to ArXiv on January 25, 2022, and located at arXiv:2201.10566, the entire contents of which are incorporated herein by reference.

[0222] exist Figure 292900. An illustrative implementation of a classic computer system 2900 that can be used in conjunction with any embodiment of the disclosure provided herein is shown in FIG. In some embodiments, any of the processes described herein can be implemented on and / or using the computer system 2900. The computer system 2900 can include one or more processors 2910 and one or more articles of manufacture, the one or more articles of manufacture including non-transitory computer-readable storage media (e.g., memory 2920 and one or more non-volatile storage media 2930). The processor 2910 can control the writing of data to the memory 2920 and the non-volatile storage device 2930 and the reading of data from the memory 2920 and the non-volatile storage device 2930 in any suitable manner. To perform any of the functions described herein, the processor 2910 can execute one or more processor-executable instructions stored in one or more non-transitory computer-readable storage media (e.g., memory 2920), which can serve as a non-transitory computer-readable storage medium for storing processor-executable instructions executed by the processor 2910.

[0223] Thus, having described several aspects and embodiments of the technology set forth in this disclosure, it will be appreciated that various variations, modifications, and improvements will readily occur to those skilled in the art. Such variations, modifications, and improvements are intended to be within the spirit and scope of the technology described herein. For example, a person of ordinary skill in the art will readily conceive of various other means and / or structures for performing the functions and / or obtaining the results and / or one or more advantages described herein, and each of such variations and / or modifications is considered to be within the scope of the embodiments described herein. Those skilled in the art will recognize or be able to determine many equivalents to the specific embodiments described herein using no more than routine experiments. Therefore, it is to be understood that the foregoing embodiments are presented only as examples, and within the scope of the appended claims and their equivalents, the inventive embodiments may be practiced in a manner different from that specifically described. In addition, any combination of two or more of the features, systems, articles, materials, equipment, and / or methods is included within the scope of this disclosure if they do not contradict each other.

[0224] The above embodiments can be implemented in any of a variety of ways. One or more aspects and embodiments of the present disclosure relating to the execution of a process or method can utilize program instructions that can be executed by a device (e.g., a computer, a processor or other device) to execute or control the execution of a process or method. In this regard, various inventive concepts can be implemented as a computer-readable storage medium (or various computer-readable storage media) (e.g., a computer memory, one or more floppy disks, compressed disks, optical disks, magnetic tapes, flash memories, circuit configurations in field programmable gate arrays or other semiconductor devices, or other tangible computer storage media) encoded with one or more programs, when executed on one or more computers or other processors, the one or more programs execute the method for implementing one or more of the above various embodiments. The computer-readable medium or medium can be removable so that one or more programs stored thereon can be loaded onto one or more different computers or other processors to implement the above various aspects. In some embodiments, the computer-readable medium can be a tangible (e.g., non-transient) computer-readable medium. In some embodiments, the computer-readable medium can include a persistent memory.

[0225] The terms "program" or "software" are used herein in a general sense to refer to any type of computer code or set of computer-executable instructions that can be used to program a computer or other processor to implement the various aspects described above. In addition, it should be recognized that, according to one aspect, one or more computer programs that, when executed, perform the methods of the present disclosure need not reside on a single computer or processor, but can be distributed in a modular manner among multiple different computers or processors to implement various aspects of the present disclosure.

[0226] Computer-executable instructions can take many forms, such as program modules, which are executed by one or more computers or other devices. Typically, program modules include routines, programs, objects, components, data structures, etc. that perform specific tasks or implement specific abstract data types. Typically, the functionality of program modules can be combined or distributed as needed in various implementations.

[0227] When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers.

[0228] Furthermore, it will be appreciated that a computer may be implemented in any of a variety of forms, such as, by way of non-limiting example, a rack-mounted computer, a desktop computer, a laptop computer, or a tablet computer. Additionally, a computer may be embedded in a device not generally considered a computer but having suitable processing capabilities, including a personal digital assistant (PDA), a smart phone, or any other suitable portable or fixed electronic device.

[0229] In addition, a computer may have one or more input devices and output devices. These devices may be used to present a user interface, etc. Examples of output devices that can be used to provide a user interface include a printer or display screen for visual presentation of output and a speaker or other sound generating device for auditory presentation of output. Examples of input devices that can be used for a user interface include a keyboard and a pointing device such as a mouse, touchpad, and digitizing tablet. As another example, a computer may receive input information through speech recognition or other audible formats.

[0230] Such computers may be interconnected in any suitable manner via one or more networks, including local area networks or wide area networks, such as enterprise networks and intelligent networks (IN) or the Internet. Such networks may be based on any suitable technology and may operate according to any suitable protocol, and may include wireless networks, wired networks, or fiber optic networks.

[0231] Furthermore, as described, some aspects can be implemented as one or more methods. The actions performed as part of a method can be ordered in any suitable manner. Thus, embodiments can be constructed in which actions are performed in an order different from that shown, and even if actions are shown as sequential in an illustrative embodiment, embodiments can include performing some actions simultaneously.

[0232] 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.

[0233] Unless explicitly stated to the contrary, the indefinite articles "a" and "an" as used in this specification and claims should be understood to mean "at least one".

[0234] The phrase "and / or" as used in this specification and claims should be understood to mean "one or both" of the elements so combined, i.e., elements that are present in combination in some cases and separately in other cases. Multiple elements listed with "and / or" should be interpreted in the same manner, i.e., "one or more" elements so combined. In addition to the elements specifically identified by the "and / or" clause, other elements may optionally be present, whether or not related to the specifically identified elements. Thus, as a non-limiting example, a reference to "A and / or B," when used in conjunction with open language such as "comprising," may refer to A alone (optionally including elements other than B) in one embodiment; to B alone (optionally including elements other than A) in another embodiment; to both A and B (optionally including other elements) in yet another embodiment; etc.

[0235] As used herein in the specification and in the claims, the phrase "at least one" with respect 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 does not necessarily include at least one element of each and every element specifically listed within the list of elements, and does not exclude any combination 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. Thus, as a non-limiting example, "at least one of A and B" (or equivalently, "at least one of A or B," or equivalently "at least one of A and / or B") may mean, in one embodiment, at least one A, optionally including more than one A, with no B (and optionally including elements other than B); in another embodiment, at least one B, optionally including more than one B, with no A (and optionally including elements other than A); in yet another embodiment, at least one A, optionally including more than one A, and at least one B, optionally including more than one B (and optionally including other elements); etc.

[0236] In the claims and the preceding description, all transitional phrases such as "including," "comprising," "with," "having," "containing," "involving," "having," "comprising," etc., shall be understood as open-ended, i.e., meaning including but not limited to. Only the transitional phrases "consisting of" and "consisting essentially of" shall be closed or semi-closed transitional phrases, respectively.

[0237] The terms "approximately" and "about" may be used to mean within ±20% of a target value in some embodiments, within ±10% of a target value in some embodiments, within ±5% of a target value in some embodiments, and within ±2% of a target value in some embodiments. The terms "approximately" and "about" may include the target value.

Claims

1. A method of operating a circuit quantum electrodynamics system, the circuit quantum electrodynamics system comprising an auxiliary qubit dispersively coupled to a first logical qubit, the method comprising: The quantum operation is performed at least in part by: generating a first drive waveform comprising a first comb of π pulses having selective frequencies corresponding to a first selection of even and odd cavity resonance frequencies of the first logical qubit and applying the first drive waveform to the auxiliary qubit; as well as The state of the auxiliary qubit is read out.

2. The method according to claim 1, further comprising: Prior to reading out the state of the auxiliary qubit, a second drive waveform is generated and applied to the auxiliary qubit, the second drive waveform comprising a second comb of π pulses having selective frequencies corresponding to a second selection of even and odd cavity resonance frequencies of the first logical qubit.

3. The method according to claim 2, wherein: The first selection includes selective frequencies 3x, 4x, 7x and 8x, and The second selection includes selective frequencies 1x, 2x, 5x and 6x.

4. The method according to claim 1, wherein The circuit quantum electrodynamics system also includes a second logical qubit coupled to the first logical qubit by a first beam splitter, and the method further includes applying a third driving waveform to the first beam splitter to perform a detuned beam splitter interaction between the first logical qubit and the second logical qubit before reading out the state of the auxiliary qubit.

5. The method according to claim 4, wherein Performing the quantum operation includes generating a Bell state between the first logical qubit and the second logical qubit.

6. The method according to claim 4, wherein: Performing the detuned beam splitter interaction between the first logical qubit and the second logical qubit includes performing the detuned beam splitter interaction between a first cavity resonator and a second cavity resonator.

7. The method according to claim 1, wherein Generating and applying the first drive waveform includes generating and applying a microwave waveform.

8. The method according to claim 1, wherein Generating and applying the first drive waveform includes generating the first drive waveform and applying the first drive waveform to a superconducting transport.

9. The method of claim 4 , further comprising generating a first four-qubit cluster state at least in part by: applying a fourth drive waveform to a second beam splitter coupling the first logical qubit and the third logical qubit; and A fifth drive waveform is applied to a third beam splitter that couples the second logical qubit to a fourth logical qubit.

10. The method of claim 9, further comprising generating a multi-qubit cluster state at least in part by: A sixth drive waveform is applied to a fourth beam splitter that couples the first logical qubit of the first four-qubit cluster state and the first logical qubit of the second four-qubit cluster state.

11. A quantum information processing system comprising: Auxiliary qubits; a first logical qubit dispersion-coupled to the ancillary qubit; as well as at least one controller configured to: The quantum operation is performed at least in part by: generating a first drive waveform comprising a first comb of π pulses having selective frequencies corresponding to a first selection of even and odd cavity resonance frequencies of the first logical qubit and applying the first drive waveform to the auxiliary qubit; as well as The state of the auxiliary qubit is read out.

12. The quantum information processing system according to claim 11, wherein: The at least one controller is further configured to generate and apply a second drive waveform to the auxiliary qubit prior to reading out the state of the auxiliary qubit, the second drive waveform comprising a second comb of π pulses having selective frequencies corresponding to a second selection of even and odd cavity resonance frequencies of the first logical qubit.

13. The quantum information processing system according to claim 12, wherein: The first selection includes selective frequencies 3x, 4x, 7x and 8x, and The second selection includes selective frequencies 1x, 2x, 5x and 6x.

14. The quantum information processing system of claim 11, further comprising a second logical qubit coupled to the first logical qubit by a beam splitter.

15. The quantum information processing system according to claim 14, wherein: The at least one controller is further configured to generate and apply a third drive waveform to the beam splitter to perform a detuned beam splitter interaction between the first logical qubit and the second logical qubit before reading out the state of the auxiliary qubit.

16. The quantum information processing system according to claim 15, wherein: The at least one controller being configured to perform the quantum operation includes the at least one controller being configured to generate a Bell state between the first logical qubit and the second logical qubit.

17. The quantum information processing system according to claim 14, wherein: The first logical qubit and the second logical qubit include a first cavity resonator and a second cavity resonator.

18. The quantum information processing system according to claim 11, wherein: The first drive waveform includes a microwave waveform.

19. The quantum information processing system according to claim 11, wherein: The auxiliary quantum bit includes a superconducting transporter.

20. A method of operating a circuit quantum electrodynamics system, the circuit quantum electrodynamics system comprising an auxiliary qubit dispersion-coupled to a first logical qubit and a second logical qubit coupled to the first logical qubit by a first beam splitter, the method comprising: Applying a first drive waveform to the auxiliary qubit, the first drive waveform comprising a π / 2 pulse; applying a second drive waveform to the first beam splitter to perform a detuned beam splitter interaction between the first logical qubit and the second logical qubit; Applying a third drive waveform to the auxiliary qubit, the third drive waveform comprising a π / 2 pulse; as well as The state of the auxiliary qubit is read out.

21. The method according to claim 20, wherein The circuit quantum electrodynamics system further includes a third logical qubit, the third logical qubit being coupled to the first logical qubit by a second beam splitter, and the method further includes: After applying the second drive waveform, a fourth drive waveform is applied to the second beam splitter to perform a detuned beam splitter interaction between the first logical qubit and the third logical qubit.

22. A method of operating a circuit quantum electrodynamics system, the circuit quantum electrodynamics system comprising a first auxiliary qubit dispersion-coupled to a first logical qubit and a second auxiliary qubit dispersion-coupled to a second logical qubit, the first logical qubit being coupled to the second logical qubit by a first beam splitter, the method comprising: applying a first drive waveform to the first beam splitter to cause a resonant beam splitter interaction between the first logical qubit and the second logical qubit; as well as Determine whether at least one of the first logical qubit and the second logical qubit is in a vacuum state by the following operation: applying a second driving waveform to the first auxiliary qubit to measure a state of the first logical qubit; as well as A third driving waveform is applied to the second auxiliary qubit to measure a state of the second logical qubit.

23. A method of operating a circuit quantum electrodynamics system, the circuit quantum electrodynamics system comprising a first auxiliary qubit dispersion-coupled to a first logical qubit, a second auxiliary qubit dispersion-coupled to a second logical qubit, and a third logical qubit, the first logical qubit and the second logical qubit being coupled by a first beam splitter, and the second logical qubit and the third logical qubit being coupled by a second beam splitter, the method comprising: preparing an arbitrary logical state in the first logical qubit; Creating a Bell state between the second logical qubit and the third logical qubit; as well as performing error correction on the arbitrary logical state by teleporting the arbitrary logical state from the first logical qubit to the third logical qubit, the teleporting comprising: introducing interference between the first logical qubit and the second logical qubit using the first beam splitter; as well as After using the first beam splitter, at least one measurement of the state of the first logical qubit and the second logical qubit is performed using the first ancillary qubit and the second ancillary qubit.

24. The method according to claim 23, wherein Preparing the Bell state comprises: preparing a first coherent state in the second logical qubit; preparing a second coherent state in the third logical qubit; and A series of joint parity measurements are performed on the second logical qubit and the third logical qubit.

25. A circuit quantum electrodynamics system comprising: Auxiliary qubits; as well as A plurality of logical qubits, the plurality of logical qubits comprising: a first logical qubit dispersion-coupled to the ancillary qubit; as well as A second logical qubit is coupled to the first logical qubit by a beam splitter.

26. The circuit quantum electrodynamics system according to claim 25, wherein: The auxiliary qubit includes a superconducting transport qubit.

27. The circuit quantum electrodynamics system according to claim 25, wherein: The second logical qubit includes a plurality of logical qubits.

28. The circuit quantum electrodynamics system according to claim 27, wherein: A logical qubit in the plurality of logical qubits comprises a bosonic mode.

29. A system comprising: The circuit quantum electrodynamics system according to claim 6; as well as at least one controller configured to: preparing an arbitrary logical state in the first logical qubit; creating a Bell state between the second logical qubit and the third logical qubit; and performing error correction on the arbitrary coherent state by teleporting the arbitrary logical state from the first logical qubit to the third logical qubit, the teleporting comprising: introducing interference between the logical qubit and the second logical qubit using at least one beam splitter; as well as After using the at least one beam splitter, at least one measurement of the state of the first logical qubit and the second logical qubit is performed using the first ancillary qubit and the second ancillary qubit.

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