Measurement-based fault-tolerant architecture for quibbler code

Abstract

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

Description

[0001] Cross-reference to related applications

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

[0003] Statement on Federally Funded Research

[0004] This invention was carried out with government support under license number W911NF-18-1-0212 granted by the U.S. Army Research Office. The government has certain rights to this invention. Background Technology

[0005] Quantum information processing techniques perform computations by manipulating one or more quantum objects. These techniques are sometimes referred to as "quantum computing." To perform computations, quantum information processors utilize quantum objects to reliably store and retrieve information. Based on some quantum information processing methods, quantum simulations of classical computational "bits" (equal to 1 or 0) have been developed, which are called qubits or "qubits." A qubit can be composed of any quantum system having two distinct states (which can be considered as a 1 state and a 0 state), but it also possesses the special property that the system can be placed in a quantum superposition and thus exist in both states simultaneously. Summary of the Invention

[0006] Some embodiments relate to a method of operating a circuit quantum electrodynamic system, the circuit quantum electrodynamic system including an auxiliary qubit dispersionally coupled to a first logic qubit. The method includes performing quantum operations at least in part by: generating a first driving waveform and applying the first driving waveform to the auxiliary qubit, the first driving waveform including a first comb of π pulses having selective frequencies corresponding to a first selection of even and odd cavity resonant frequencies of the first logic qubit; and reading out the state of the auxiliary qubit.

[0007] Some implementations relate to a quantum information processing system comprising: an auxiliary qubit; a first logic qubit dispersionally coupled to the auxiliary qubit; and at least one controller configured to perform quantum operations at least in part by: generating a first driving waveform and applying the first driving waveform to the auxiliary qubit, the first driving waveform comprising a first comb of π pulses having selective frequencies corresponding to a first selection of even and odd cavity resonant frequencies of the first logic qubit; and reading out the state of the auxiliary qubit.

[0008] In some implementations, the method includes generating a second driving waveform and applying the second driving waveform to the auxiliary qubit before reading out the state of the auxiliary qubit. The second driving waveform includes a second comb of π pulses having selective frequencies corresponding to a second selection of even and odd cavity resonant frequencies of the first logic qubit.

[0009] In some implementations, 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 logic qubit coupled to the first logic qubit by the first beam splitter, and the method further includes applying a third driving waveform to the first beam splitter before reading out the state of the auxiliary qubit to enable detuned beam splitter interaction between the first logic qubit and the second logic qubit.

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

[0012] In some implementations, detuned beamsplitter interaction between the first logic qubit and the second logic qubit includes detuned beamsplitter interaction between the first cavity resonator and the second cavity resonator.

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

[0014] In some implementations, generating and applying the first driving waveform includes generating the first driving waveform and applying the first driving waveform to the superconducting transporter.

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

[0016] In some embodiments, the method further includes generating a multi-qubit cluster state at least in part by applying a sixth driving 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 electrodynamic system, the circuit quantum electrodynamic system including an auxiliary qubit dispersionally coupled to a first logic qubit and a second logic qubit coupled to the first logic qubit by a first beam splitter. The method includes: applying a first driving waveform to the auxiliary qubit, the first driving waveform including a π / 2 pulse; applying a second driving waveform to the first beam splitter to perform detuned beam splitter interaction between the first logic qubit and the second logic qubit; applying a third driving waveform to the auxiliary qubit, the third driving waveform including a π / 2 pulse; and reading out the state of the auxiliary qubit.

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

[0019] Some embodiments relate to a method for operating a circuit quantum electrodynamic system, the circuit quantum electrodynamic system including a first auxiliary qubit dispersionally coupled to a first logic qubit and a second auxiliary qubit dispersionally coupled to a second logic qubit, the first logic qubit being coupled to the second logic qubit by a first beam splitter. The method includes: applying a first driving waveform to the first beam splitter to induce resonant beam splitter interaction between the first and second logic qubits; and determining whether at least one of the first and second logic qubits is in a vacuum state by applying a second driving waveform to the first auxiliary qubit to measure the state of the first logic qubit, and applying a third driving waveform to the second auxiliary qubit to measure the state of the second logic qubit.

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

[0021] In some implementations, preparing a 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 and third logical qubits.

[0022] Some implementations relate to a circuit quantum electrodynamics system comprising: an auxiliary qubit; and a plurality of logic qubits, the plurality of logic qubits including a first logic qubit dispersionally coupled to the auxiliary qubit, and a second logic qubit coupled to the first logic qubit by a beam splitter.

[0023] In some implementations, the auxiliary qubits include superconducting transporter qubits.

[0024] In some implementations, the second logical qubit comprises multiple logical qubits.

[0025] In some implementations, the logical qubits in a plurality of logical qubits include boson modes.

[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 the arbitrary coherent state by teleporting the arbitrary logical state from the first logical qubit to the third logical qubit, the teleportation comprising: introducing interference between the logical qubit and the second logical qubit using at least one beam splitter; and after using at least one beam splitter, performing at least one measurement of the states of the first logical qubit and the second logical qubit using a first auxiliary qubit and a second auxiliary qubit. Attached Figure Description

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

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

[0029] Figure 2 This is a schematic diagram of another illustrative quantum information processing system based on some embodiments of the technology described herein.

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

[0031] Figure 3B Some embodiments of the technology described herein can be used to implement Figure 3A A schematic diagram illustrating a quantum information processing system using quantum circuitry.

[0032] Figure 3C This is a schematic diagram of an illustrative quantum circuit for performing parity measurements according to some embodiments of the technology described herein.

[0033] Figure 3D It is for performing some implementations of the technology described herein. Figure 3C A schematic diagram of the illustrative driving waveform for parity measurement.

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

[0035] Figure 4B This is a schematic diagram of an illustrative quantum circuit for performing Z-measurement according to some embodiments of the technology described herein.

[0036] Figure 4C It is for performing some implementations of the technology described herein. Figure 4B A schematic diagram of the illustrative driving waveform for Z-measurement.

[0037] Figure 5 This is a schematic diagram of an illustrative quantum circuit for performing Z-based fault-tolerant measurements according to some embodiments of the technology described herein.

[0038] Figure 6 This is a schematic diagram of an illustrative quantum circuit for performing X-based fault-tolerant measurements according to some embodiments of the technology described herein.

[0039] Figure 7 This is a schematic diagram of an illustrative quantum circuit for performing XX-based fault-tolerant measurements according to some embodiments of the technology described herein.

[0040] Figure 8A This is a schematic diagram of an illustrative quantum circuit for performing ZZ-based fault-tolerant measurements according to some embodiments of the technology described herein.

[0041] Figure 8B This is for implementing some embodiments of the technology described herein. Figure 8A A schematic diagram illustrating a quantum information processing system using quantum circuitry.

[0042] Figure 8C This is a schematic diagram of an illustrative quantum circuit for performing ZZ measurements according to some embodiments of the technology described herein.

[0043] Figure 8D It is for performing some implementations of the technology described herein. Figure 8B A schematic diagram of the illustrative driving waveform for ZZ measurement.

[0044] Figure 9A This is a schematic diagram of an illustrative quantum circuit for performing ZZZ-based fault-tolerant measurements according to some embodiments of the technology described herein.

[0045] Figure 9B This is for implementing some embodiments of the technology described herein. Figure 9A A schematic diagram illustrating a quantum information processing system using quantum circuitry.

[0046] Figure 9C This is a schematic diagram of an illustrative quantum circuit for performing ZZZ measurements according to some embodiments of the technology described herein.

[0047] Figure 9D It is for performing some implementations of the technology described herein. Figure 9CA schematic diagram of the illustrative driving waveform for ZZZ measurement.

[0048] Figure 10 This is a flowchart of a process 1000 for performing quantum operations according to some embodiments of the technology described herein.

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

[0050] Figure 12 This is a schematic diagram of an illustrative quantum circuit for performing telecorrection according to some embodiments of the technology described herein.

[0051] Figure 13 This 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 This 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 This is a schematic diagram of an illustrative quantum circuit for preparing the |χ> state according to some embodiments of the technology described herein.

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

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

[0056] Figure 17B These are some implementations of the technology described herein. Figure 17A A detailed schematic diagram of the quantum circuit.

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

[0058] Figure 19 This is a schematic diagram of an illustrative quantum circuit for performing a SWAP test between a first qubit and a second qubit, according to some embodiments of the technology described herein.

[0059] Figure 20 This 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 This is a schematic diagram illustrating the effects of the Kerr effect and χ′ on quantum states according to some embodiments of the technology described herein.

[0061] Figure 22A This is a graph illustrating examples of drive waveforms generated using a frequency comb according to some embodiments of the technology described herein.

[0062] Figure 22B This illustrates some embodiments based on the technology described herein. Figure 22A A graph of the Fourier transform of the driving waveform.

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

[0064] Figure 23B This illustrates some embodiments based on the technology described herein. Figure 23A The Fourier transform curve of the driving waveform.

[0065] Figure 24 This is a flowchart describing another process 2400 for performing quantum operations according to some embodiments of the technology described herein.

[0066] Figure 25A This is a schematic diagram of another illustrative quantum circuit configured to prepare a Bell state with two qubits, according to some embodiments of the technology described herein.

[0067] Figure 25B Some embodiments of the technology described herein can be used Figure 25A A schematic diagram of a two-qubit ZZ Bell state cluster state prepared by quantum circuitry.

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

[0069] Figure 26B Some embodiments of the technology described herein can be used Figure 26A A schematic diagram of a four-qubit cluster state prepared by quantum circuitry.

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

[0071] Figure 27B Some embodiments of the technology described herein can be used Figure 27A A schematic diagram of a two-qubit entangled state prepared by quantum circuitry.

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

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

[0074] Figure 29 This is an illustrative schematic diagram of a conventional computer system according to some embodiments of the technology described herein. Detailed Implementation

[0075] Several different types of qubits have been successfully demonstrated in the laboratory. However, the lifetime of the states in many of these systems is currently around 100 μs before information loss occurs due to decoherence of the quantum state or other quantum noise. Despite this long lifetime, error correction techniques are crucial for enabling reliable storage and retrieval of information stored in quantum systems. However, unlike conventional computing systems where bits can be copied for error correction purposes, cloning the unknown states of a quantum system is impossible. Nevertheless, the system can be entangled with other quantum systems that efficiently propagate information from the system across several entangled objects.

[0076] This application relates to an improved quantum error correction technique for correcting errors in the state of a quantum system exhibiting one or more boson modes. In this context, "error" refers to a change in the state of a quantum system that may be caused by, for example, boson loss, boson gain, dephase, or time evolution of the system, and an "error" that alters the state of the system changes the information stored in the system.

[0077] As mentioned above, quantum multilevel systems, such as qubits, exhibit quantum states that decoherent within approximately 100 μs based on current experimental practice. Therefore, coupling a multilevel system with another system exhibiting a longer decoherence time may be beneficial. As will be described below, boson modes are particularly desirable for coupling to multilevel systems. Through this coupling, the state of the multilevel system can be alternatively represented by boson modes, thus retaining the same information in the state for a longer period compared to when they exist alone in the multilevel system in other ways.

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

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

[0080] When a multi-level system, such as a qubit, is mapped to the state of a bosonic system coupled to it, 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 represent the ground state of a qubit using the zero-boson-number state of the resonator, and the excited state of the qubit using the one-boson-number state of the resonator. That is:

[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 amplitudes of the qubit in state |g> or |e>, respectively, and |0> and |1> are the zero-boson-number state and the one-boson-number state of the resonator, respectively. While this is perfectly valid code, it is not robust to many errors, such as boson loss. That is, when boson loss occurs, this code may not be able to recover the resonator's state before the boson loss.

[0084] The code usage can be written more generally as follows:

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

[0087] When an error occurs, the system state changes to the resulting state, which we will refer to as the "error word" in this paper.

[0089] Here, index k refers to the specific error that has occurred. As mentioned above, examples of errors include boson loss, boson gain, dephase, amplitude decay, etc. Generally, the choice of code affects the system's robustness to errors. That is, when an error occurs, the code used determines the extent to which the previous state can be faithfully recovered. An ideal code would be associated with a wide range of error types so that when any error occurs, information is not lost and any quantum superposition of the logical codewords can be faithfully recovered.

[0090] However, a challenge of the above approach is that the code may be limited by the lifetime of the nonlinear auxiliary qubits required for quantum control of the boson system. Typically, the boson system is controlled, and errors in the boson system are corrected by manipulating auxiliary qubits coupled to it. However, this can mean that error correction of the boson system's state may no longer be possible when an error occurs in the auxiliary qubits.

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

[0092] The inventors have developed a general set of operations for fault-tolerant parity operations based on boson systems. Specifically, they have extended the use of fault-tolerant parity measurements, enabling non-destructive and fault-tolerant measurements of Z, ZZ, and ZZZ logical operators in basic code. Implementations of these logical operators involve measuring them by performing detuned beamsplitter interactions while the auxiliary is in a superposition state. In some implementations, the ZZ and ZZZ operators can be measured even when the auxiliary is directly coupled to only one of a plurality of logical qubits.

[0093] Using fault-tolerant parity measurements and extensions discussed above, the inventors have also developed methods for preparing Z and X eigenstates, Bell states, and GHZ states in quadruple code. Furthermore, the inventors have developed methods for performing robust measurements with Z, X, ZZ, and XX logic bases by combining beam splitter and cavity photon number measurements. For example, the implementation of the X measurement utilizes beam splitter interactions, using coherent states to interfere with the logic state. Subsequently, a photon number-selective driving waveform is applied to the auxiliary qubit to determine whether one of the logic qubits (e.g., the cavity) is in a vacuum state. These measurements are fault-tolerant to superconducting transporter decay and dephase errors of all orders in the sense that overall measurement errors can be suppressed exponentially by repeated measurements and majority voting on the results.

[0094] The inventors also recognized and understood that, combined with cavity shift operations, this set of operators is sufficient to perform Clifford operations in rudimentary code while maintaining first-order fault tolerance to quantum errors. To make this set general, the inventors developed operations including fault-tolerant SNAP gates to achieve arbitrary single-qubit Z-rotations, or alternatively, operations to prepare high-fidelity arbitrary states on a single-qubit Bloch sphere via a distillation scheme. This involves generating N incomplete copies of the target state and comparing the copies pairwise by performing a non-destructive fault-tolerant SWAP test among all possible copy pairs. A post-selection is performed after all SWAP tests have passed, resulting in N copies of the target state with higher fidelity than the initial state.

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

[0096] According to some implementations, the code described herein can be used to configure the state of a boson system. A boson system can be a particularly ideal system in which the techniques described herein are applied, because individual boson modes can exhibit equidistant spacing of coherent states. For example, a resonator cavity is a simple harmonic oscillator with equidistant horizontal spacing. Boson modes also facilitate quantum communication because they can be stationary for quantum memories, or facilitate 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 is depicted, applicable to various aspects of practicing this application. In system 100, quantum system 101 includes an auxiliary qubit 110 coupled to logic qubit 120 via dispersive coupling. That is, the detuning of the auxiliary qubit to the logic qubit is much greater (e.g., an order of magnitude greater) than the coupling strength between auxiliary qubit 110 and logic qubit 120. Logic qubit 120 is also coupled to logic qubit 140 by a beam splitter 130 (e.g., a programmable beam splitter). Energy source 150 can provide energy to one or two of the auxiliary qubit 110, logic qubit 120, beam splitter 130, and / or logic qubit 140 to perform operations on the system, such as preparing a state in any of the logic qubits 120 and / or 140, measuring one or more of the logic qubits 120 and / or 140, applying gate operations to one or more of the logic qubits 120 and / or 140, applying operations to the auxiliary qubit 110 or preparing a state in the auxiliary qubit 110, detecting and / or correcting errors in the auxiliary qubits 110 and / or the logic qubits 120 and / or 140, or a combination thereof.

[0099] According to some implementations, logic qubits 120 and 140 can be implemented as any suitable multimode boson system. While this may include photonic systems, such as one or more microwave cavities, the techniques described herein are not limited to such systems. Logic qubits 120 and 140 can be implemented as multimode boson systems, which may include multiple modes of a single boson system and / or any combination of single modes of multiple boson 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, quantum systems based on superconducting Josephson junctions, such as charge qubits (Cooper-pair boxes), flux qubits or phase qubits, superconducting transport qubits, or combinations thereof. Auxiliary qubit 110 may be coupled to logic qubit 120 via dispersion coupling, which couples the state of auxiliary qubit 110 to the state of logic qubit 120. Logic qubit 120 may comprise any boson system supporting multiple boson 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, logic qubit 120 may comprise multiple transmission line resonators.

[0101] According to some implementations, beam splitter 130 can be configured to provide switchable beam splitter interaction between logic qubit 120 and one or more logic qubits 140. For example, each beam splitter 130 can be actuated between one of logic qubit 120 and logic qubit 140 in the form of...

[0102] System 100 also includes a power 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-computed driving waveforms may be stored on the storage medium 170 and accessed by the controller 160 to apply the waveforms to subsystem 101. For example, the controller 160 may (e.g., in response to user input provided to the controller) access the driving waveforms 172 stored on the storage medium 170 and subsequently control the power source 150 to apply one or more driving 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 can also be referred to as “driving” the auxiliary qubit and / or logic qubit. Coupling can be achieved using any technique, such as coupling the electric and / or magnetic fields generated by the auxiliary and logic qubits. According to some embodiments, the auxiliary qubit (e.g., a superconducting transporter) can be coupled to the logic qubit, which acts as a mechanical resonator, via piezoelectric coupling. According to some embodiments, the auxiliary qubit can be coupled to the logic qubit, which acts as a magnetic resonator, by coupling the auxiliary qubit (e.g., a superconducting transporter) to a phonon—which in turn is coupled to a magnon via magnetostrictive coupling.

[0104] Figure 2 Alternative illustrative systems suitable for practicing various aspects of this application are described. In system 200, quantum system 201 includes auxiliary qubits 110 coupled to logic qubits 140 via dispersive coupling. Logic qubits 140 are also coupled to other logic qubits 140 by beam splitters 130. Beam splitters 130 are switchable to turn beam splitter interactions between any pair of logic qubits 140 on and off. Energy source 150 can provide energy to one or both of the auxiliary qubits 110, beam splitters 130, and / or logic qubits 140 to perform operations on the system, such as preparing states in any logic qubit 140, measuring the states of one or more logic qubits 140, applying gate operations to one or more logic qubits 140, applying operations to the auxiliary qubits 110, detecting and correcting errors in the auxiliary qubits 110 and / or logic qubits 140, or combinations thereof.

[0105] II. Operations for Four-Legged Cat Code

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

[0107] To use this encoding as a quantum memory, logical states need to be prepared in appropriate codewords, single-photon loss errors need to be detected and corrected, and then the logical information needs to be read out from the quantum system. To further utilize this encoding for quantum computing, a universal gate set must be additionally implemented.

[0108] Without quantum control of a simple harmonic oscillator, neither quantum memory nor computation would be possible. To achieve quantum control of a simple harmonic oscillator using classical external actuation, nonlinear sources can be added to the system. For example, auxiliary qubits (such as superconducting transport qubits) can be added to the system, discretely coupled to the simple harmonic oscillator (e.g., a microwave cavity resonator). Unfortunately, these auxiliary qubits can be additional sources of error that can propagate into the information stored in the simple harmonic oscillator.

[0109] Because these errors generated by the auxiliary qubits possess quantum properties, they can be described as transition operators. Although the possible quantum errors are infinite, correcting the most probable errors in this cavity-superconducting transport subsystem within the time window between error correction steps significantly improves computational performance. Such errors include single-photon losses in simple harmonic oscillators, single decays of excitations in the auxiliary qubits, and / or dephasing of states stored in the auxiliary qubits. This set of errors can be concisely summarized as follows:

[0111] Where |g> and |e> are the first two stages of the auxiliary qubit, and

[0113] Here, |f> is the third stage of the auxiliary qubit.

[0114] If the suboperations described herein are designed so that they do not cause logical errors in the qubits of a simple harmonic oscillator when one of the aforementioned errors occurs, then the operations are fault-tolerant to these errors. For this condition to be met, either the error can be corrected at a later time, or the impact of the error on the logical information stored in the simple harmonic oscillator is negligible.

[0115] The inventors have recognized and understood that the level of fault tolerance required for universal quantum computing using basic quantum code 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 proceeds by preparing qubits in entangled resource states, including many-body entangled states, also known as "cluster states." Computation can then be performed using cluster states by measuring qubits in a specific cardinality. Quantum operations do not directly implement logic gates but can be divided into the preparation of quantum states and destructive measurements; these operations are then used to implement quantum gates and quantum error correction.

[0116] The first quantum operation for implementing fault-tolerant quantum computing in rudimentary code is a simple harmonic oscillator (e.g., combined with the one described in this article). Figure 1 or Figure 2 The state preparation in the aforementioned logical qubits (120 or 140). Figure 3A This is a schematic diagram of an illustrative quantum circuit 300 for fault-tolerant fabrication of the |+> state in a qubit according to some embodiments of the technology described herein.

[0117] In some implementations, the quantum circuit 300 describes the operations applied to a single qubit in a left-to-right reading order. At the leftmost position, the qubit begins in a vacuum state (|vac>). Subsequently, a shift 302 (D(α)) can be applied to shift the qubit's state to a coherent state (e.g., α = 2 to 3). After shifting the qubit's state, a parity measurement 304 can be repeatedly performed. Fault tolerance for auxiliary errors is achieved by requiring repeated parity measurements to yield the same result. If each parity measurement 304 yields a different result, an error is inferred, and the state can be discarded. By requiring two measurements to be consistent, a state |α>±|-α> that is fault-tolerant to a set of errors can be prepared.

[0118] In some implementations, the following methods can be used: Figure 3B The illustrated quantum information processing system 310 implements the quantum circuit 300. The quantum information processing system 310 includes a logic qubit 312 depicted as a microwave cavity resonator. The logic qubit 312 is dispersion-coupled to an auxiliary superconducting transport sub-qubit 314. A readout resonator 316 (e.g., a microwave strip resonator) is coupled to the auxiliary superconducting transport sub-qubit 314 and is configured to provide input to and / or read out information from the auxiliary superconducting transport sub-qubit 314.

[0119] In some implementations, such as Figure 3C As depicted in the example, the parity measurement 304 can be described as a sequence of quantum operations applied to auxiliary qubits and logic qubits. Figure 3C In quantum circuits, the auxiliary qubit |g> is depicted within the logical qubit |ψ>. L >The row below. Quantum operations include the first π / 2 rotation of the auxiliary qubit 304a, the unitary operation applied to the logic qubit 304b, the second π / 2 rotation of the auxiliary qubit 304c, and the measurement of the state of the auxiliary qubit 304d.

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

[0121] Figure 4A This describes another quantum operation for state preparation performed in "four-legged cat code". Figure 4A In the example, according to some implementations, the quantum circuit 400 is configured to prepare a four-legged cat state |a>±|iα>+|a>±|-iα>, which serves as the logic 0 and 1 codewords for the four-legged cat code. The quantum circuit 400 starts from the logical qubit being in a vacuum state (|vac>). Thereafter, as in combination Figure 3A As mentioned above, displacement 302(D(α)) can be applied. To prepare the four-legged cat state in a fault-tolerant manner, it can be done according to... Figure 4A The sequence described applies a series of parity measurements 304 and logic Z measurements 406.

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

[0123] In some implementations, the logic Z-measure 406 can be described as a sequence of quantum operations applied to the auxiliary qubit and the logic qubit, such as Figure 4B The example depicts this. The only difference between logic Z-measure 406 and parity measure 304 is the execution wait time (T) of unitary operation 406b. 4Π =π / 2χ=T Π / 2) is related to combination Figure 3B Half of the unitary operation 304b. Similarly, as... Figure 4C The series of driving waveforms 410 depicted are 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, it is also necessary to measure the states of logical qubits as part of the implementation of quantum computing. Figure 5 This is a schematic diagram of a quantum circuit 500 according to some embodiments of the technology described herein, which is used to perform fault-tolerant measurements in a Z-based system of basic code. The quantum circuit 500 includes logic qubits |ψ L Measurement 502. This measurement 502 can be destructive because if a decay error occurs during measurement 502, it will cause the state stored in the logical qubit to dephase. In this case, although the state stored in the logical qubit cannot be used for further logical operations thereafter, measurements can continue to improve the overall measurement fidelity through a majority vote of repeated measurement results.

[0125] In some implementations, measurement 502 in the Z-basis of the four-legged code can be physically implemented by applying optimized control pulses to the auxiliary qubit to excite the auxiliary qubit only if the logic qubit contains photons of n = 0, 3, 4, 7, 8, ... . Alternatively, measurement 502 can be implemented by driving the auxiliary qubit with a linear combination of selective π pulses of appropriate frequencies.

[0126] Figure 6 This is a schematic diagram of an illustrative quantum circuit 600 according to some embodiments, which is used to perform fault-tolerant measurements in a base of four-legged cat code. To perform fault-tolerant measurements in the X base, the quantum circuit 600 includes the use of auxiliary logic qubits, as shown by the lower line of the circuit diagram. The auxiliary logic qubits can start from a vacuum state (|vac>) and can subsequently be shifted by a displacement 602 (D(α)) (e.g., as in combination with...). Figure 3C The displacement 302 described above is prepared in a coherent state. This is achieved through the coherent state and the logical quantum bit |ψ LBeam splitter interference 604 between the states, measurements in the X basis distinguish between the |α>±|-α> state and the |iα>±|-iα> state. Measurements 606 and 608 (e.g., implemented using selective π pulses) determine whether only one of the logical qubits and auxiliary qubits contains a 0 photon. If exactly one of the logical qubits and auxiliary qubits contains a 0 photon, then the input state is |α>±|-α>, because only when the input state is |iα>±|-iα>, there is a 0 photon.

[0127] Figure 7 This is a schematic diagram of an illustrative quantum circuit 700 according to some embodiments, which is used to perform fault-tolerant measurements in a base of XX-based code. Fault-tolerant measurements and combinations in the XX-based code. Figure 6 The measurement described in 600 is very similar. The measurement in quantum circuit 700 does not use logic qubits and auxiliary qubits, but rather originates from two logic qubits |ψ L1 > and |ψ L2 >To begin. If, by measuring 606 and 608, one of the logic qubits is found to contain 0 photons, this indicates that the two cats are aligned in the same direction in phase space.

[0128] Figure 8A This is a schematic diagram of an illustrative quantum circuit 800 according to some embodiments, which is used to perform fault-tolerant measurements in a ZZ-based four-legged code. In some embodiments, the quantum circuit 800 includes a measurement 802 of the joint 4-parity of two logic qubits.

[0129] One way to measure the joint 4-parity of two logic qubits is to couple an auxiliary qubit to a single-cavity mode stored in the logic qubit. Then, a single-mode 4-parity measurement sequence can be performed without measuring any states. Then, as in conjunction with the [document / parameter / section]... Figure 19 The process involves applying a SWAP operation followed by another single-mode 4 parity measurement. Afterward, auxiliary qubits can be measured in the X basis, and another SWAP operation can be performed. However, this process is limited by the fact that the beam splitter rate is typically less than X, necessitating the elimination of X during the SWAP operation.

[0130] A faster sequence to avoid this problem is to combine the SWAP operation and the dispersive Hamiltonian into a single operation that achieves joint 4-parity measurement. To understand how it works, it's important to first note that the joint 4-parity operator jointly rotates the cavity phase space by 90 degrees:

[0132] or equivalent to:

[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. Using the correct timing and the ratio of χ to g, any unitary element can be generated from the Hamiltonian:

[0136] For a given value of χ, the two first operation points are set to

[0137] These specific ratios can achieve the desired element. This measurement can be combined with... Figures 3A to 3D The method for fault-tolerant parity measurement is similar to that described above, providing fault tolerance for superconducting transporter errors. When using the χ-matched measurement |e>, the gf manifold of the auxiliary qubit allows for the detection of superconducting transporter decay errors. Even if a single superconducting transporter decays, this measurement will not cause the cavity to dephase. Photon losses are correctable, provided that the parity is tracked using the parity measurement and the 4-parity measurement is updated accordingly before the next parity transition occurs. 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 last π / 2 pulse.

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

[0139] In some implementations, the quantum circuit 800 can use Figure 8B The illustrated quantum information processing system 810 is used to implement this. The quantum information processing system 810 includes a first logic qubit 812a and a second logic qubit 812b. Both the first logic qubit 812a and the second logic qubit 812b can be microwave cavity resonators, such as... Figure 8B The example depicts a first logic qubit 812a and a second logic qubit 812b coupled together by a beam splitter 814. The first logic qubit 812a is dispersively 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 and / or read out information from the auxiliary superconducting transport sub-qubit 816.

[0140] In some implementations, the joint 4-parity measurement 802 of the 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. In Figure 8C In quantum circuits, operations on auxiliary qubits |g> are described in logic qubits |ψ>. L1 > and |ψ L2 >The row below. Storage status | ψ L1 The logical qubit is a logical qubit that is dispersively coupled to an auxiliary qubit, and the logical qubit |ψ L1 > and |ψ L2 Coupled by a beam splitter, such as in combination Figure 8B As illustrated in the example. The quantum operation includes: a first π / 2 rotation 802a of the auxiliary qubit, applied to the logic qubit |ψ L1 > and |ψ L2 > Beam splitter operation 802b, second -π / 2 rotation of auxiliary qubit 802c, and measurement of state of auxiliary qubit 802d.

[0141] like Figure 8D As shown in the example, Figure 8C The quantum operations can be physically implemented by applying a series of 820 driving waveforms to the auxiliary qubit. This series of 820 includes: a first sequence 822a, which includes driving waveforms comprising g-eπ / 2 pulses and e-fπ pulses; and a subsequent second sequence 822b, which also includes driving waveforms comprising e-fπ pulses and g-eπ / 2 pulses. Sequences 822a and 822b are time-delayed.

[0143] For time T ZZ ,in

[0144] Figure 9A This is a schematic diagram of an illustrative quantum circuit 900 according to some embodiments, used to perform fault-tolerant measurements in a ZZZ basis of cat-and-mouse code. Quantum circuit 900 can be viewed as an extension of quantum circuit 800 and includes 3-qubit measurements instead of 2-qubit measurements. To achieve ZZ measurements in quantum circuit 800, a switchable beamsplitter interaction between a1 and a2 and a third-level auxiliary qubit dispersively coupled to a1 are used. To extend this to ZZZ measurements in quantum circuit 900, an additional switchable beamsplitter 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:

[0146] The two pairwise beam splitter interactions, sequentially implemented between the first and second logical qubits, and then between the first and third logical qubits, and then with appropriate waiting time, can be used to perform the aforementioned unitary element.

[0147] Performing the interaction of these two consecutive paired beam splitters yields the following unitary element:

[0149] This equation shows that the first logical qubit has accumulated an additional conditional phase. If a measurement is performed after these two beam splitter interactions, the operator Π1Z2Z3, where Π is the parity of the photon number of the first logical qubit, is used. To counteract this, the waiting time T = π / (2χ) produces the following unitary element:

[0151] After reorganization, we can conclude that:

[0153] Return to Figure 9A In some implementations, the quantum circuit 900 can use Figure 9B The illustrated quantum information processing system 910 is used to implement this. The quantum information processing system 910 includes a first logical qubit 912a, a second logical qubit 912b, and a third logical qubit 912c. For example... Figure 9B In the example depicted, all three qubits 912a, 912b, and / or 912c can be microwave cavity resonators. The first logic qubit 912a and the second logic qubit 912b are coupled to each other by a beam splitter 914a. The first logic qubit 912a and the third logic qubit 912c are coupled to each other via another beam splitter 914b. The first logic qubit 912a is dispersively coupled to an auxiliary superconducting transport subqubit 916. A readout resonator 918 (e.g., a microwave strip resonator) is coupled to the auxiliary superconducting transport subqubit 916 and is configured to provide input to and / or read out information from the auxiliary superconducting transport subqubit 916.

[0154] In some implementations, 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. In Figure 9C In quantum circuits, operations on auxiliary qubits |g> are described in logic qubits |ψ>. L1 >、|ψ L2 > and |ψ L3 >On the line below. Storage status | ψ L1The logical qubits of > are logical qubits that are dispersively coupled to auxiliary qubits, and the paired logical qubits |ψ L1 > and |ψ L2 >and|ψ L1 > and |ψ L3 Each free beam splitter is coupled, such as in combination Figure 9B As described in the example. The quantum operations include: a first π / 2 rotation 902a of the auxiliary qubit; and the application of the quantum to the logic qubit |ψ L1 > and |ψ L2 > Beam splitter operation 902b; applied to logic qubits |ψ L1 > and |ψ L3 The second beam splitter operation 902c; applied to the first logic qubit |ψ L1 The unitary operation 902d; the second -π / 2 rotation of the auxiliary qubit 902e; and the measurement of the state of the auxiliary qubit 902f.

[0155] like Figure 9D As shown in the example, Figure 9C The quantum operations can be physically implemented by applying a series of 920 driving waveforms to an auxiliary qubit and a beam splitter. This series 920 includes: a first sequence 922a, which includes driving waveforms comprising g-eπ / 2 pulses and e-fπ pulses; and a subsequent second sequence 922b, which also includes driving waveforms comprising e-fπ pulses and g-eπ / 2 pulses. Sequences 922a and 922b are time-delayed by 2T. ZZ +T 4Π Interval. Delay by 2T at this time. ZZ +T 4Π During this process, a driving waveform is applied to the two beamsplitters of the coupled logical qubits to induce detuned beamsplitter interaction, which has a Hamiltonian of the form described above. The length is T. 4Π The time period can be used to correct any rotation of the state stored in the already accumulated logical qubits (e.g., -90°, -45°, and -45° for the first, second, and third logical qubits, respectively).

[0156] Figures 9A to 9CThe ZZZ measurement completes the Clifford gate set of the four-legged code because it can be used to implement CNOT gates when combined with the other quantum operations mentioned above. Entangled states |+++>+|---> can be created by using the ZZZ measurement on the separable state |+++>. Similar entangled states |000>+|111> can be created by measuring pairs of ZZ operators (e.g., Z1Z2 and Z2Z3) on the same initial state |+++>. Belli measurements on these states can deterministically implement CNOT gates until local Pauli corrections are made, thus forming a gate set that is known to be universal.

[0157] Figure 10 This is a flowchart of a process 1000 for performing quantum operations according to some embodiments described herein. Process 1000 can be used to operate, for example, a quantum information processing system including circuit quantum electrodynamic components. The quantum information processing system may include auxiliary qubits (such as superconducting transport qubits, SNAILmon qubits, oscillators, or other qubits) coupled to a first logic qubit (such as a microwave cavity resonator). The first logic qubit can be coupled to a second logic qubit via a first beam splitter.

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

[0159] In some embodiments, process 1000 may begin at action 1010, wherein a first driving waveform may be applied to the auxiliary qubit. The first driving waveform may include a π / 2 pulse. In some embodiments, the first driving waveform may include a sequence of driving waveforms. For example, a series of driving waveforms may include g-eπ / 2 pulses and e-fπ pulses.

[0160] In some implementations, after action 1010, process 1000 can proceed to action 1020. In action 1020, a second driving waveform can be applied to the first beam splitter to induce detuned beam splitter interaction between the first logic qubit and the second logic qubit. The detuned beam splitter interaction can...

[0162] For time T ZZ ,in

[0163] In some embodiments, after action 1020, process 1000 may proceed to action 1030, in which a third driving waveform may be applied to the auxiliary qubit. The third driving waveform may include a π / 2 pulse. In some embodiments, the first driving waveform may include a sequence of driving waveforms. For example, the sequence of driving waveforms may include e-fπ pulses and g-eπ / 2 pulses.

[0164] In some embodiments, after action 1030, process 1000 can proceed to action 1040, in which the state of the auxiliary qubit can be read out. In some embodiments, the state of the auxiliary qubit can 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, the state of the auxiliary qubit can be measured. For example, the state of the auxiliary qubit can be destructively measured. In some embodiments, this measurement can be performed, for example, using 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 can be a null 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 in two steps. First, an entangled Bell pair can be created. Second, a measurement is performed on one half of the Bell pair in a Bell basis with the unknown state. Until a known Pauli correction (depending on the measurement result) is achieved, the unknown state is deterministically transferred to the other half of the Bell pair after that measurement.

[0167] A long-standing and prominent problem with four-legged cat code is the so-called "no-leap" reaction, which causes the cat's "size" α to decrease over time. This no-leap reaction can be mitigated by teleporting the quantum information to a new logical code with a larger α if the correct Bell state can be created. For example, if the first logical qubit starts at 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 the effective size.

[0168] By generating Bell states between α = α′ qubits in the logic basis and α = α0 qubits in the logic basis, suitable Bell states can be created to correct the no-transition reaction. Figure 11 The 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 logic qubits. The first logic qubit can be shifted by a shift of 1102 (D1(α)), and the second logic qubit can be shifted by a shift of 1104 (D2(β)), thereby preparing two quantum states in the first and second logic qubits. In some embodiments, the first and second logic qubits can initialize the states in different logic bases. Figure 11 In the example, 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 two logical qubits in different logical bases makes it possible to correct the no-transition reaction.

[0170] Subsequently, if combined Figures 3A to 3D The parity measurement 304 can be performed twice, once for the first logic qubit and once for the second logic qubit. Then, the quantum circuit 1100 can continue to perform two consecutive ZZ measurements, such as in combination. Figures 8A to 9C As described above. Subsequently, two additional parity measurements 304 can be performed on each of the first and second logical qubits. As described herein, to ensure fault-tolerant generation of Bell states, the first and second parity measurements 304 must be consistent for each of the first and second logical qubits. 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 implicit correction, such as Figure 12 This is depicted in quantum circuits. First, the first qubit of Bell state 1100 (prepared with a cat code of size α) can be connected to the logic qubit |ψ L A beam splitter interaction 1202 is performed between >_α. Then, both the first qubit and the logic qubit of the Bell state 1100 can be measured using measurement 1204 in the Z-basis of the four-legged code. In some embodiments, measurement 1204 can be equivalent to the combination herein. Figure 5 The aforementioned measurement 502. Subsequently, both the first qubit and the logic qubit of Bell state 1100 can be measured using measurement 1206 in the XX basis of the four-legged cat code, which can be equivalent to the measurement combined with the method described herein. Figure 6 The aforementioned measurement 606. These measurements then transfer the data originally stored in the logical qubit |ψ L The quantum information in the > is teleported to the second qubit of the Bell state 1100 (with a cat code of size β), thereby correcting the no-jump reaction and preventing leakage errors from accumulating in multiple quantum operations.

[0172] Since the beam splitter preserves the parity of the total photon number (i.e., preserves the photon number), the ZZ information can still be extracted by measuring the local photon number parity mod(4) and summing the results. The protocol is fault-tolerant because after the beam splitter, all logical XX and ZZ information has been mapped to the cavity's non-local photon number space. Although auxiliary qubit errors can still cause logical qubits to dephase during the teleportation correction process, at least two photon losses are required in either cavity to produce incorrect measurement results.

[0173] In particular, for cluster state models in quantum computing, a potentially useful subroutine is to create Greenberger-Horne-Zeilinger (GHZ) entangled states, such as |000>+|111> and |+++>+|--->. Figure 13 This is a schematic diagram of an illustrative quantum circuit 1300 for preparing |000>+|111> GHz cluster states according to some embodiments described herein. The 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. Subsequently, a parity measurement 304 is performed on each logical qubit. A first pair of ZZ measurements 802 is performed on the first and second qubits, and subsequently a second pair of ZZ measurements 802 is performed on the second and third qubits. Finally, a parity measurement 304 is performed on each logical qubit. As previously stated, to provide fault tolerance, the first and last parity measurements must be consistent, the first pair of ZZ measurements 802 must be consistent, and the second pair of ZZ measurements 802 must be consistent to prepare the |000>+|111> GHz cluster states.

[0174] Figure 14 This is a schematic diagram of another illustrative quantum circuit 1400 for preparing |+++>+|--->GHZ cluster states according to some embodiments described herein. Quantum circuit 1400 is similar to... Figure 13 The quantum circuit 1300, however, does not perform two pairs of ZZ measurements 802 between the two sets of parity measurements 304, but rather a pair of ZZZ measurements 902. 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, and then performing a Bell measurement between that entangled state and the logical qubit to simultaneously teleport information to the remaining unmeasured qubits and execute the gate. The gate can be executed until some Pauli correction is achieved (depending on the measurement results).

[0176] To prepare a CNOT gate, we can first prepare a |χ> state, such as Figure 15 The example of quantum circuit 1500 is depicted. The |χ> state can be described as...

[0177] According to some implementations, once the |χ> state is prepared, it can be used for the stealth teleportation CNOT gate, such as... Figure 16 The quantum circuit 1600 is shown. A CNOT gate can be teleported using Bell measurements 1200 between pairs of logic qubits 1 and 2 and logic qubits 5 and 6. The output of the 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 should be recognized that there may be more efficient ways to compile quantum circuits with CNOT gates, thereby reducing the number of operations and measurements, but this explicit construction is very useful for demonstrating that the set of operations described herein is indeed universal.

[0178] Combining state preparation with Bell state measurement also enables the execution of a fault-tolerant Hadamard gate between two logical qubits. Figure 17A This is based on some implementation methods for preparing hadamard state |Φ Had A schematic diagram of the quantum circuit 1700. Figure 17B An extended version of the quantum circuit 1700 is depicted in the text.

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

[0180] Subsequently, a destructive measurement is performed on one of the qubits in the X basis using measurement 600, which utilizes an additional auxiliary qubit initialized in a different logic basis than the three logical qubits. Measurement 600 includes a beam splitter interaction 1708 between one of the logical qubits and the auxiliary qubit, and then a destructive measurement of the states of the logical qubit and the auxiliary qubit is performed using measurement 606. The destructive measurement on this logical qubit in the X basis projects the biqubit states of the other two logical qubits onto the state |+i+i>±i|-ii>, where the sign is determined by both the ZZZ measurement 902 and the result of the X measurement 600.

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

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

[0183] IV. State purification using the SWAP test

[0184] Purification via SWAP testing is a general method used to symmetrize ordinary qubits. The inventors have recognized and appreciated that this method can be used to prepare states in boson qubits with high fidelity. Specifically, when generating several copies of a target state using an error-prone (e.g., noisy) procedure, a non-destructive SWAP test between pairs of states can be used to reduce errors. The SWAP test results can then be post-selected to further reduce errors in state generation.

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

[0186] The SWAP test is fault-tolerant to first-order errors, and therefore the initial state preparation error can be much larger than the SWAP test error. Under these conditions, the SWAP test can be used to purify the initial state and reduce state preparation errors. The process begins by preparing N noisy copies of the desired quantum state to be initialized. For simplicity, we can assume that the probability of some errors occurring in state preparation is p. err The probability of not making an error is (1-p) err When a SWAP test measurement is performed between two of these cavities, the measurement result indicates the probability p of failure. err / 2 is very small, so the protocol must be restarted. However, in most cases, the SWAP test measurement will succeed, producing two logical qubits with an error probability of p. err / 2. This direct trade-off between success probability and state fidelity is highly advantageous.

[0187] By repeatedly performing SWAP tests on all different paired cavities, the error probability can be reduced as the SWAP test measurement succeeds, until the limit set by the fidelity measured by the SWAP test is reached. Since SWAP testing can be performed fault-tolerantly, 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 general operation is described. Figure 19This is a schematic diagram of an illustrative quantum circuit 1900 according to some embodiments, which is used to fault-tolerantly implement a SWAP test between a first qubit and a second qubit. In this context, the SWAP test is a non-destructive measurement of the SWAP operator between two logical qubits. If |ψ1> and |ψ2> are the initial input states, the SWAP test will project these states onto...

[0189] To perform this measurement, a 50-50 beamsplitter interaction is first performed between the two logical qubits 1900a. Then, within the beamsplitter framework, the photon number parity is measured using one of the 304 measurement modes for parity measurement. Typically, parity measurement of a single logical qubit involves measuring the parity operator.

[0190] In terms of face value, the results of the SWAP test are quite easy to interpret. If the result is +1 (i.e., the auxiliary 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 performing a post-selection on this result, the probability of any state being erroneous 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 all possible errors, it is possible that |<ψ1|ψ2>| = 0. Furthermore, obtaining a result of +1 does not guarantee that an error did not occur. If |<ψ1|ψ2>| = 0, it is still possible to obtain a result of +1 with a probability of 0.5. Therefore, when the SWAP test passes, the error in both cavity states is halved, but never completely eliminated.

[0192] This can be expressed more precisely in density matrix form. The initial noise 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 |wherein,|ψ t >The target state is to be prepared with high fidelity, and |ψ i > represents the state obtained when an error occurs during the initial preparation, where <ψ i |ψ t >=0, and p i It is a real scalar that sums to 1.

[0194] The initial dual-cavity state can be written as

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

[0196] Figure 20 This is a schematic diagram of an illustrative quantum circuit 2000 according to some embodiments of the technology described herein, configured to reduce errors present in quantum states prepared in four qubits. The quantum circuit 2000 includes several SWAP tests 1900 and SWAP operations 2002 between paired cavities. Figure 20 In the example, the process begins at ρ init Four copies. During the implementation of Quantum Circuit 2000, a SWAP test 1900 can be performed between all six permutations of paired logical qubits to reduce the error of each state to p. err / 8. Using ρ init Adding an additional copy can further reduce the error rate, at the cost of adding more swap tests and swap operations. (See the attached copy in this article.) Figures 8A to 8D The protocol can be experimentally implemented using the same hardware as ZZ measurements.

[0197] V. Corrections for the Kerr effect and χ'

[0198] In some quantum information processing schemes, it is desirable to account for additional effects that may introduce perturbations into the quantum system. For example, the Kerr effect and the χ' effect can affect the ZZ and / or ZZZ measurements described herein, making them less robust. These effects are particularly pronounced for systems utilizing 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 qubits. For example, the χ' effect is quadratically proportional to the number of photons stored in the logical qubits, making it more difficult to distinguish and correct the χ' effect the larger the number of photons. Consideration of these effects is especially important in MBQC, where an even larger number of photons are used to perform computational processes.

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

[0201] Figure 21The diagram illustrates a Bloch sphere according to some embodiments, showing the effects of the Kerr effect and χ' on the quantum state. 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 the quantum circuit.

[0202] To counteract these effects, a substitution process can be used to prepare the quantum state stored in the logical qubit, and similarly, the quantum operations can be altered. In some implementations, the cat state can be prepared by first shifting the state of the logical qubit from the vacuum state |vac> to the state |α>. Subsequently, the |α> state can be driven to |0> using a driving waveform comprising a selective g-fπ pulse comb. L The selective g-fπ pulse comb can consist of multiple π pulses corresponding to frequencies (0χ, 4χ, 8χ, 12χ, ...). Using these selective frequencies in the g-fπ pulse comb resolves the influence of χ', while changing the phase of the component π pulses resolves perturbations of the Kerr effect, as these phases provide a secondary correction to the equidistant spacing of the logic qubit energy levels. Figure 22A An example of a selective g-fπ pulse comb is shown in the figure. Figure 22B The corresponding Fourier spectrum is shown in the figure.

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

[0204] Figure 24This is a flowchart describing another process 2400 for performing quantum operations according to some embodiments of the technology described herein. Process 2400 can be used to operate, for example, a quantum information processing system including circuit quantum electrodynamic components. The quantum information processing system may include auxiliary qubits (such as superconducting transporter qubits, SNAILmon qubits, oscillators, or other qubits) coupled to a first logic qubit (such as a microwave cavity resonator).

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

[0206] In some embodiments, the first driving waveform includes a first comb of π pulses having selective frequencies corresponding to a first selection of even and odd cavity resonant frequencies of the first logic qubit. For example, the first comb of π pulses may have selective frequencies corresponding to frequencies of (3χ, 4χ, 7χ, 8χ, ...).

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

[0208] In some embodiments, after action 2410 or 2420, process 2400 may proceed to action 2440, in which the state of the auxiliary qubit can be read out. In some embodiments, the state of the auxiliary qubit can 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, the state of the auxiliary qubit can be measured. For example, the state of the auxiliary qubit can be destructively measured. In some embodiments, such a measurement can be performed, for example, using 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 null 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 electrodynamic system further includes a second logical qubit coupled to the first logical qubit via a first beamsplitter. For example, the first and second logical qubits may each be microwave cavity resonators coupled to the first beamsplitter. The method may include applying a third driving waveform to the first beamsplitter before reading out the state of the auxiliary qubit to induce a detuned beamsplitter interaction between the first and second logical qubits. Thereafter, process 2400 may proceed to the 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 driving waveform to a second beam splitter coupling the first and third logical qubits to enable beam splitter interaction between the first and third logical qubits. Furthermore, the four-qubit cluster state can be generated by applying a fifth driving waveform to a third beam splitter coupling the second logical qubit to the fourth logical qubit. In this way, quantum states stored in the four logical qubits can be entangled to create a 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 may be the XZZX cluster state described herein, or any other multi-qubit cluster state suitable for MBQC. The multi-qubit cluster state may be generated at least in part by applying a sixth driving waveform to a fourth beam splitter coupling the first logical qubit of the first four-qubit cluster state and the first logical qubit of the second four-qubit cluster state.

[0212] VI. Cluster preparation

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

[0214] Figure 25A This is a schematic diagram of an illustrative quantum circuit 2500 according to some embodiments, which is configured to prepare Bell states in two qubits. Figure 25B This is a schematic diagram of a two-qubit ZZ Bell state 2510. In some embodiments, a two-qubit ZZ Bell state 2510 can be prepared using a quantum circuit 2500. Figure 25BThe illustration includes two qubits 2512 fabricated in the first logic basis, represented by closed circles. The lines connecting the two qubits 2512 represent entanglement coupling.

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

[0216] like Figure 26A As shown, an example of a four-qubit cluster state can be created through the interaction of a chain beam splitter. Figure 26B This 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 started in

[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 logic basis. Figure 27A This is a schematic diagram of a quantum circuit 2700 according to some embodiments, which is configured to generate Figure 27B The depicted two-qubit entangled state 2710 includes a first qubit 2512 prepared in a first logic basis (e.g., X) and a second qubit 2714 prepared in a second different logic basis (e.g., Z).

[0218] In some implementations, the quantum circuit 2700 is used in

[0219] Figure 28A This is a schematic diagram illustrating a fusion process according to some embodiments of the technology described herein, which can be used to generate another four-qubit cluster state. The process may begin at stage 2800, where four separate cluster states comprise three two-qubit states and one four-qubit state. A Bell measurement 2802 (which may 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 binding) 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 may include the XZZX cluster states described herein, or alternatively or additionally include RHG cluster states. Figure 28B This 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, four-qubit cluster states 2610 and 2810 can be fused to form a larger cluster state, such as cluster state 2820. These larger cluster states can be further fused to create a final cluster state for MBQC or other applications. Figure 28B As depicted in the paper, in some embodiments, the larger cluster state can be the XZZX cluster state 2830. Further 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, at arXiv:2201.10566, the entire contents of which are incorporated herein by reference.

[0222] exist Figure 29 This document illustrates an illustrative implementation of a classic computer system 2900 that can be used in conjunction with any of the embodiments disclosed herein. In some embodiments, any of the processes described herein may be implemented on and / or using the computer system 2900. The computer system 2900 may include one or more processors 2910 and one or more articles of manufacture, one or more of which include non-transitory computer-readable storage media (e.g., memory 2920 and one or more non-volatile storage media 2930). The processor 2910 may control the writing of data to and from the memory 2920 and non-volatile storage media 2930 in any suitable manner. To perform any of the functions described herein, the processor 2910 may execute one or more processor-executable instructions stored in one or more non-transitory computer-readable storage media (e.g., memory 2920), which may be used as a non-transitory computer-readable storage medium for storing the processor-executable instructions executed by the processor 2910.

[0223] Therefore, having described several aspects and implementations of the technology set forth in this disclosure, it should be recognized that various changes, modifications, and improvements will readily occur to those skilled in the art. Such changes, modifications, and improvements are intended to remain within the spirit and scope of the technology described herein. For example, those skilled in the art will readily conceive of various other means and / or structures for performing 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 implementations described herein. Those skilled in the art will recognize or be able to determine many equivalents of the particular implementations described herein using experiments not exceeding conventional methods. Therefore, it is to be understood that the foregoing embodiments are presented by way of example only, and that inventive embodiments may be practiced in ways other than those specifically described within the scope of the appended claims and their equivalents. Furthermore, any combination of two or more of the features, systems, articles, materials, apparatuses, and / or methods is included within the scope of this disclosure unless they contradict each other.

[0224] The above-described embodiments can be implemented in any of a variety of ways. One or more aspects and embodiments of this disclosure relating to the execution of processes or methods can be executed or controlled using program instructions executable by means of an apparatus (e.g., a computer, processor, or other means). In this regard, various inventive concepts can be implemented as a computer-readable storage medium (or various computer-readable storage media) (e.g., computer memory, one or more floppy disks, compact disks, optical disks, magnetic tapes, flash memory, field-programmable gate arrays or other semiconductor devices, or other tangible computer storage media) encoding one or more programs that, when executed on one or more computers or other processors, perform methods implementing one or more of the various embodiments described above. The computer-readable medium or medium can be portable, such that one or more programs stored thereon can be loaded onto one or more different computers or other processors to implement the various aspects described above. In some embodiments, the computer-readable medium can be a tangible (e.g., non-transitory) computer-readable medium. In some embodiments, the computer-readable medium can include persistent memory.

[0225] Throughout this document, the terms "program" or "software" are used 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. Furthermore, it should be understood that, according to one aspect, one or more computer programs that perform the methods of this disclosure during execution do not need to reside on a single computer or processor, but can be distributed in a modular manner across multiple different computers or processors to implement the various aspects of this disclosure.

[0226] Computer-executable instructions can take many forms, such as program modules 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. The functionality of a program module can usually be combined or distributed in various implementations as needed.

[0227] When implemented in software, the software code can be executed on any suitable processor or set of processors, whether it is set up in a single computer or distributed across multiple computers.

[0228] Furthermore, it should be recognized that, by way of non-limiting example, a computer can be implemented in any of a variety of forms, such as a rack-mount computer, a desktop computer, a laptop computer, or a tablet computer. Additionally, a computer can be embedded in a device that is not typically considered a computer but has suitable processing power, including a personal digital assistant (PDA), a smartphone, or any other suitable portable or stationary electronic device.

[0229] In addition, a computer may have one or more input and output devices. These devices can 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 pointing devices such as a mouse, touchpad, and digitizer. As another example, a computer may receive input information through speech recognition or other audible formats.

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

[0231] Furthermore, as described, some aspects can be implemented as one or more methods. Actions performed as part of a method can be ordered in any suitable manner. Therefore, implementations can be constructed that perform actions in an order different from the one shown; even actions shown as sequential in the illustrative implementation may include actions performed simultaneously.

[0232] All definitions used and defined herein should be understood to be governed by the dictionary definition, the definition in references incorporated herein by reference, and / or the general meaning of the terms defined.

[0233] Unless expressly stated to the contrary, the indefinite articles “a” and “an” used in this specification and claims shall 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 two” of the elements so combined—that is, elements that exist together in some cases and separately in others. Multiple elements listed with “and / or” should be interpreted in the same way, 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 they are related to the specifically identified elements. Thus, as a non-limiting example, a reference to “A and / or B” used in conjunction with open-ended language such as “comprising” may, in one embodiment, refer only to A (optionally including elements other than B); in another embodiment, refer only to B (optionally including elements other than A); in yet another embodiment, refer to A and B (optionally including other elements); and so on.

[0235] As used herein in the specification and claims, the phrase "at least one" relating to a list of one or more elements should be understood to mean at least one element selected from any one or more elements in the list, but does not necessarily include every single element specifically listed in the list and at least one element from each element, and does not exclude any combination of elements in the list. This definition also allows for the optional presence of elements other than those specifically identified in the list of elements referred to by the phrase "at least one," whether related to or unrelated to those specifically identified elements. Therefore, 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") can refer to at least one A in one embodiment, optionally including more than one A, with no B (and optionally including elements other than B); at least one B in another embodiment, optionally including more than one B, with no A (and optionally including elements other than A); at least one A in yet another embodiment, optionally including more than one A, and at least one B, optionally including more than one B (and optionally including other elements); and so on.

[0236] In the claims and the above description, all transitional phrases such as "comprising," "including," "with," "having," "containing," "involving," "holding," "comprising," etc., should be understood as open-ended, meaning including but not limited to. Only the transitional phrases "composed of" and "substantially composed of" should be closed or semi-closed transitional phrases, respectively.

[0237] The terms "approximately" and "about" may be used in some embodiments to mean within ±20% of the target value, in some embodiments to mean within ±10% of the target value, in some embodiments to mean within ±5% of the target value, and in some embodiments to mean within ±2% of the target value. 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 ancilla qubit dispersively coupled to a first logical qubit, the method comprising: performing a quantum operation at least in part by: generating and applying to the ancilla qubit a first drive waveform, the first drive waveform comprising 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 ancilla qubit. generating and applying to the ancilla qubit a second drive waveform prior to reading out the state of the ancilla qubit, the second drive waveform comprising a second comb of pi pulses having selective frequencies corresponding to a second selection of even and odd cavity resonance frequencies of the first logical qubit.

2. The method of claim 1, further comprising:

3. The method of claim 2, wherein: the first selection comprises selective frequencies 3x, 4x, 7x, and 8x, and the second selection comprises selective frequencies lx, 2x, 5x, and 6x. the circuit quantum electrodynamics system further comprises a second logical qubit coupled to the first logical qubit by a first beamsplitter, the method further comprising applying a third drive waveform to the first beamsplitter to perform a detuned beamsplitter interaction between the first logical qubit and the second logical qubit prior to reading out the state of the ancilla qubit.

4. The method of claim 1, wherein, performing the quantum operation comprises generating a Bell state between the first logical qubit and the second logical qubit.

5. The method of claim 4, wherein, performing the detuned beamsplitter interaction between the first logical qubit and the second logical qubit comprises performing the detuned beamsplitter interaction between a first cavity resonator and a second cavity resonator.

6. The method of claim 4, wherein, generating and applying the first drive waveform comprises generating and applying a microwave waveform.

7. The method of claim 1, wherein, generating and applying the first drive waveform comprises generating the first drive waveform and applying the first drive waveform to a superconducting transmon.

8. The method of claim 1, wherein, 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 beamsplitter coupling the first logical qubit and a third logical qubit; and applying a fifth drive waveform to a third beamsplitter coupling 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: applying a sixth drive waveform to a fourth beamsplitter coupling a first logical qubit of the first four-qubit cluster state and a first logical qubit of a second four-qubit cluster state.

11. A quantum information processing system comprising: an ancilla qubit; a first logical qubit dispersively coupled to the ancilla qubit; and at least one controller configured to: perform a quantum operation at least in part by: ​ generating a first drive waveform and applying the first drive waveform to the ancilla qubit, the first drive waveform comprising 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 ancilla qubit.

12. The quantum information processing system of claim 11, wherein, the at least one controller is further configured to, prior to reading out the state of the ancilla qubit, generate a second drive waveform and apply the second drive waveform to the ancilla qubit, the second drive waveform comprising a second comb of pi 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 of claim 12, wherein: the first selection comprises selective frequencies 3x, 4x, 7x, and 8x, and the second selection comprises selective frequencies lx, 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 beamsplitter.

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

16. The quantum information processing system of claim 15, wherein, the at least one controller being configured to perform the quantum operation comprises 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 of claim 14, wherein, the first logical qubit and the second logical qubit comprise first and second cavity resonators.

18. The quantum information processing system of claim 11, wherein, the first drive waveform comprises a microwave waveform.

19. The quantum information processing system of claim 11, wherein, the ancilla qubit comprises a superconducting transmon.

20. A method of operating a circuit quantum electrodynamics system comprising an ancilla qubit dispersively coupled to a first logical qubit and a second logical qubit coupled to the first logical qubit by a first beamsplitter, the method comprising: applying a first drive waveform to the ancilla qubit, the first drive waveform comprising pi / 2 pulses; applying a second drive waveform to the first beamsplitter to perform a detuned beamsplitter interaction between the first logical qubit and the second logical qubit; applying a third drive waveform to the ancilla qubit, the third drive waveform comprising pi / 2 pulses; and reading out a state of the ancilla qubit.

21. The method of claim 20, wherein, the circuit quantum electrodynamics system further comprises a third logical qubit coupled to the first logical qubit by a second beamsplitter, and the method further comprises: after applying the second drive waveform, applying a fourth drive waveform to the second beamsplitter to perform a detuned beamsplitter 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 ancilla qubit dispersively coupled to a first logical qubit and a second ancilla qubit dispersively coupled to a second logical qubit, the first logical qubit coupled to the second logical qubit by a first beamsplitter, the method comprising: applying a first drive waveform to the first beamsplitter to perform a resonant beamsplitter 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 ancilla qubit to measure a state of the first logical qubit; and applying a third drive waveform to the second ancilla 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 ancilla qubit dispersively coupled to a first logical qubit, a second ancilla qubit dispersively coupled to a second logical qubit, and a third logical qubit, the first logical qubit and the second logical qubit coupled by a first beamsplitter and the second logical qubit and the third logical qubit coupled by a second beamsplitter, the method comprising: 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 comprising: introducing an interference between the first logical qubit and the second logical qubit using the first beamsplitter; and performing at least one measurement of a state of the first logical qubit and the second logical qubit using the first ancilla qubit and the second ancilla qubit after using the first beamsplitter.

24. The method of 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 performing a series of joint parity measurements on the second logical qubit and the third logical qubit.

25. A circuit quantum electrodynamics system comprising: an ancilla qubit; and a plurality of logical qubits, the plurality of logical qubits comprising: a first logical qubit dispersively coupled to the ancilla qubit; and a second logical qubit coupled to the first logical qubit by a beamsplitter.

26. The circuit quantum electrodynamics system of claim 25, wherein, The ancilla qubit comprises a superconducting anyon qubit.

27. The circuit quantum electrodynamics system of claim 25, wherein, The second logical qubit comprises a plurality of logical qubits.

28. The circuit quantum electrodynamics system of 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 of claim 6; and at least one controller, the at least one controller configured to: 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 coherent state by teleporting the arbitrary logical state from the first logical qubit to the third logical qubit, the teleportation comprising: introducing an 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 and second logical qubits using the first and second ancillary qubits after using the at least one beam splitter.

Images