Measurement system fault-tolerant architecture for the four-leg cat code.

Abstract

Systems and methods are provided for performing fault-tolerant quantum operations for a four-leg Cat code. The quantum system includes an ancilla qubit dispersively coupled to a first logical qubit, and the quantum system may be manipulated at least in part by: generating and applying a first drive waveform to the ancilla qubit, where the first drive waveform includes a first comb of 7t pulses having selective frequencies corresponding to a first selection of even and odd cavity resonant frequencies of the first logical qubit; and reading out the state of the ancilla qubit.

Description

[Technical field]

[0001] REFERENCE TO RELATED APPLICATIONS This application claims the benefit under 35 USC § 119(e) of U.S. Provisional Patent Application No. 63 / 293,034, filed December 22, 2021, and entitled "MEASUREMENT-BASED FAULT TOLERANT ARCHITECTURE FOR THE 4-LEGGED CAT CODE," which is incorporated by reference in its entirety.

[0002] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH This invention was made with Government support under W911NF-18-1-0212 awarded by the United States Army Research Office. The Government has certain rights in the invention. [Background technology]

[0003] background 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. Some quantum information processing approaches have developed a quantum analog to classical computing "bits" (equal to 1 or 0), called quantum bits or "qubits." Qubits can consist of any quantum system that has two distinct states (which may be thought of as 1 and 0 states), but also has the special property that the system can be placed in a quantum superposition, thereby existing in both of these states at once. Summary of the Invention

[0004] Quick Overview Some embodiments relate to a method of operating a circuit quantum electrodynamic system including an ancilla qubit dispersively coupled to a first logical qubit, the method including, at least in part: generating and applying a first drive waveform to the ancilla qubit, where the first drive waveform includes a first comb of π pulses having selective frequencies corresponding to first selections of even and odd cavity resonant frequencies of the first logical qubit; and performing a quantum operation by reading out the state of the ancilla qubit.

[0005] Some embodiments relate to a quantum information processing system that includes an ancilla qubit; a first logical qubit dispersively coupled to the ancilla qubit; and at least one controller configured to generate and apply a first drive waveform to the ancilla qubit, where the first drive 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 logical qubit; and perform a quantum operation by reading out the state of the ancilla qubit.

[0006] In some embodiments, the method includes generating and applying a second drive waveform to the ancilla qubit prior to reading out the state of the ancilla qubit, where the second drive 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 logical qubit.

[0007] In some embodiments, the first selection comprises the selective frequencies 3χ, 4χ, 7χ and 8χ, and the second selection comprises the selective frequencies 1χ, 2χ, 5χ and 6χ.

[0008] In some embodiments, the circuit quantum electrodynamics system further includes a second logical qubit coupled to the first logical qubit by the first beam splitter, and the method further includes applying a third drive waveform to the first beam splitter to enact a detuned beam splitter interaction between the first logical qubit and the second logical qubit prior to reading out the state of the ancilla qubit.

[0009] In some embodiments, performing a quantum operation includes generating a Bell state between the first logical qubit and the second logical qubit.

[0010] In some embodiments, defining a detuned beam splitter interaction between the first logical qubit and the second logical qubit comprises defining a detuned beam splitter interaction between the first resonant cavity and the second resonant cavity.

[0011] In some embodiments, generating and applying the first drive waveform includes generating and applying a microwave waveform.

[0012] In some embodiments, generating and applying the first drive waveform comprises generating a first drive waveform and applying it to a transmon.

[0013] In some embodiments, the method further includes generating the first four-qubit cluster state, at least in part, by: applying a fourth drive waveform to a second beam splitter that couples the first logical qubit and the third logical qubit; and applying a fifth drive waveform to a third beam splitter that couples the second logical qubit to the fourth logical qubit.

[0014] In some embodiments, the method further includes generating a multi-qubit cluster state, at least in part, by: applying a sixth drive waveform to a fourth beam splitter that couples a first logical qubit of the first four-qubit cluster state and a first logical qubit of the second four-qubit cluster state.

[0015] Some embodiments relate to a method of operating a circuit quantum electrodynamic system including an ancilla qubit dispersively coupled to a first logical qubit and a second logical qubit coupled to the first logical qubit by a first beam splitter, the method including: applying a first drive waveform to the ancilla qubit, where the first drive waveform includes a π / 2 pulse; applying a second drive waveform to the first beam splitter to define a detuned beam splitter interaction between the first logical qubit and the second logical qubit; applying a third drive waveform to the ancilla qubit, where the third drive waveform includes a π / 2 pulse; and reading out a state of the ancilla qubit.

[0016] In some embodiments, the circuit quantum electrodynamics system further includes a third logical qubit coupled to the first logical qubit by a second beam splitter, and the method further includes: after applying the second drive waveform, applying a fourth drive waveform to the second beam splitter to define a detuned beam splitter interaction between the first logical qubit and the third logical qubit.

[0017] Some embodiments relate to a method of operating a circuit quantum electrodynamic system including 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 being coupled to the second logical qubit by a first beam splitter. The method includes: applying a first drive waveform to the first beam splitter to define an on-resonance beam splitter interaction between the first logical qubit and the second logical qubit; and 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, thereby determining whether at least one of the first and second logical qubits is in a vacuum state.

[0018] Some embodiments relate to methods of operating a circuit quantum electrodynamics system including 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, where the first logical qubit and the second logical qubit are coupled by a first beam splitter and the second logical qubit and the third logical qubit are coupled by a second beam splitter. The method includes: preparing an arbitrary logical state in a first logical qubit; preparing a Bell state between a second logical qubit and a 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, where teleporting includes: using a first beam splitter to introduce interference between the first logical qubit and the second logical qubit; and performing at least one measurement of states of the first logical qubit and the second logical qubit using the first ancilla qubit and the second ancilla qubit after using the first beam splitter.

[0019] In some embodiments, preparing the Bell state includes: 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 concatenated parity measurements on the second logical qubit and the third logical qubit.

[0020] Some embodiments relate to a circuit quantum electrodynamic system that includes a plurality of logical qubits, including an ancilla qubit; and a first logical qubit that is dispersively coupled to the ancilla qubit; and a second logical qubit that is coupled to the first logical qubit by a beam splitter.

[0021] In some embodiments, the ancilla qubit comprises a transmon qubit.

[0022] In some embodiments, the second logical qubit comprises a plurality of logical qubits.

[0023] In some embodiments, a logical qubit of the plurality of logical qubits comprises a boson mode.

[0024] In some embodiments, the system further includes at least one controller configured to: prepare an arbitrary logical state in the first logical qubit; prepare a Bell state between the second logical qubit and the 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, where teleporting includes introducing interference between the logical qubit and the second logical qubit using at least one beam splitter; and performing at least one measurement of the states of the first logical qubit and the second logical qubit using the first ancilla qubit and the second ancilla qubit after using the at least one beam splitter. [Brief description of the drawings]

[0025] BRIEF DESCRIPTION OF THE DRAWINGS Various aspects and embodiments are described with reference to the following drawings. The figures are not necessarily drawn to scale. For clarity, not all components may be labeled in every figure. In the drawings: [Figure 1] FIG. 1 is a schematic diagram of an exemplary quantum information processing system in accordance with some aspects of the technology described herein. [Diagram 2] FIG. 2 is a schematic diagram of another exemplary quantum information processing system in accordance with some aspects of the technology described herein. [Figure 3-1]3A is a schematic diagram of an example quantum circuit for fault-tolerant preparation of |+> states in a qubit according to some embodiments of the technology described herein. FIG. 3B is a schematic diagram of an example quantum information processing system that can be used to implement the quantum circuit of FIG. 3A according to some embodiments of the technology described herein. [Figure 3-2] 3C is a schematic diagram of an example quantum circuit for performing a parity measurement according to some aspects of the technology described herein. FIG. 3D is a schematic diagram of an example drive waveform for performing the parity measurement of FIG. 3C according to some aspects of the technology described herein. [Figure 4] 4A is a schematic diagram of an example quantum circuit for fault-tolerant preparation of |0> or |1> states in a qubit according to some embodiments of the technology described herein. FIG. 4B is a schematic diagram of an example quantum circuit for performing a Z measurement according to some embodiments of the technology described herein. FIG. 4C is a schematic diagram of an example drive waveform for performing the Z measurement of FIG. 4B according to some embodiments of the technology described herein. [Diagram 5] FIG. 5 is a schematic diagram of an example quantum circuit for performing fault-tolerant measurements in the Z-basis in accordance with some aspects of the technology described herein. [Figure 6] FIG. 6 is a schematic diagram of an example quantum circuit for performing fault-tolerant measurements in the X-basis in accordance with some aspects of the technology described herein. [Figure 7] FIG. 7 is a schematic diagram of an example quantum circuit for performing fault-tolerant measurements in the XX standard in accordance with some aspects of the technology described herein. [Figure 8-1] FIG. 8A is a schematic diagram of an example quantum circuit for performing fault-tolerant measurements in the ZZ standard in accordance with some aspects of the technology described herein. [Figure 8-2]Figure 8B is a schematic diagram of an example quantum information processing system that can be used to perform the quantum circuit of Figure 8A according to some aspects of the technology described herein. Figure 8C is a schematic diagram of an example quantum circuit for performing a ZZ measurement according to some aspects of the technology described herein. Figure 8D is a schematic diagram of an example drive waveform for performing the ZZ measurement of Figure 8B according to some aspects of the technology described herein. [Figure 9-1] 9A is a schematic diagram of an example quantum circuit for performing fault-tolerant measurements in the ZZZ standard according to some aspects of the technology described herein. FIG. 9B is a schematic diagram of an example quantum information processing system that can be used to perform the quantum circuit of FIG. 9A according to some aspects of the technology described herein. [Figure 9-2] 9C is a schematic diagram of an example quantum circuit for performing a ZZZ measurement according to some aspects of the technology described herein. FIG. 9D is a schematic diagram of an example drive waveform for performing the ZZZ measurement of FIG. 9C according to some aspects of the technology described herein. [Figure 10] FIG. 10 is a flowchart describing a process 1000 for performing quantum operations according to some aspects of the technology described herein. [Figure 11] FIG. 11 is a schematic diagram of an exemplary quantum circuit for preparing a Bell state in accordance with some aspects of the technology described herein. [Figure 12] FIG. 12 is a schematic diagram of an example quantum circuit for performing telecorrection in accordance with some aspects of the technology described herein. [Figure 13] FIG. 13 is a schematic diagram of an exemplary quantum circuit for preparing Greenberger-Horne-Zeilinger (GHZ) cluster states in accordance with some aspects of the technology described herein. [Figure 14]FIG. 14 is a schematic diagram of another exemplary quantum circuit for preparing a GHZ cluster state in accordance with some aspects of the technology described herein. [Figure 15] FIG. 15 is a schematic diagram of an exemplary quantum circuit for preparing a |χ〉 state in accordance with some aspects of the technology described herein. [Figure 16] FIG. 16 is a schematic diagram of an example quantum circuit for teleporting a CNOT gate, according to certain embodiments described herein. [Figure 17-1] FIG. 17A is a schematic diagram of a simplified quantum circuit for preparing a |ΦHad> state in accordance with some embodiments of the technology described herein. [Figure 17-2] FIG. 17B is a detailed schematic diagram of the quantum circuit of FIG. 17A in accordance with some aspects of the technology described herein. [Figure 18] FIG. 18 is a schematic diagram of a quantum circuit configured to teleport a Hadamard gate in accordance with some aspects of the technology described herein. [Figure 19] FIG. 19 is a schematic diagram of an example quantum circuit for fault-tolerant implementation of a SWAP test between a first and second qubit in accordance with certain aspects of the technology described herein. [Figure 20] FIG. 20 is a schematic diagram of an exemplary quantum circuit configured to reduce errors present in a quantum state prepared in four qubits, in accordance with some aspects of the technology described herein. [Figure 21] FIG. 21 is a schematic diagram illustrating the Kerr effect and the effect of χ′ on a quantum state, according to some embodiments of the technology described herein. [Figure 22] 22A is a plot illustrating an example of a drive waveform generated using a frequency comb according to some aspects of the technology described herein, and FIG 22B is a plot illustrating a Fourier transform of the drive waveform of FIG 22A according to some aspects of the technology described herein. [Diagram 23] 23A is a plot illustrating another example of a drive waveform generated using a frequency comb according to some aspects of the technology described herein, and FIG 23B is a plot illustrating a Fourier transform of the drive waveform of FIG 23A according to some aspects of the technology described herein. [Figure 24] FIG. 24 is a flowchart describing another process 2400 for performing quantum operations according to some aspects of the technology described herein. [Diagram 25] Figure 25A is a schematic diagram of another exemplary quantum circuit configured to prepare Bell states in two qubits according to some aspects of the technology described herein. Figure 25B is a schematic diagram of a two-qubit ZZ Bell state cluster state that can be prepared using the quantum circuit of Figure 25A according to some aspects of the technology described herein. [Figure 26] Figure 26A is a schematic diagram of another exemplary quantum circuit configured to prepare a four-qubit cluster state according to some aspects of the technology described herein. Figure 26B is a schematic diagram of a four-qubit cluster state that can be prepared using the quantum circuit of Figure 26A according to some aspects of the technology described herein. [Figure 27] Figure 27A is a schematic diagram of another quantum circuit configured to generate a two-qubit entangled state according to some aspects of the technology described herein. Figure 27B is a schematic diagram of a two-qubit entangled state that can be prepared using the quantum circuit of Figure 27A according to some aspects of the technology described herein. [Figure 28] Figure 28A is a schematic diagram illustrating another process for generating another four-qubit cluster state according to some embodiments of the technology described herein. Figure 28B is a schematic diagram illustrating the formation of an XZZX cluster state according to some embodiments of the technology described herein. [Figure 29]FIG. 29 is a schematic diagram of an exemplary conventional computer system in accordance with some embodiments of the technology described herein. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0026] Detailed Description Several different kinds of qubits have been successfully demonstrated in the laboratory. However, the lifetime of many states of these systems before information is lost due to quantum state decoherence or other quantum noise is currently about 100 μs. Despite the longer lifetimes, it may be important to provide error correction techniques in quantum computing that allow reliable storage and recovery of information stored in quantum systems. However, unlike classical computing systems where bits can be copied for error correction purposes, it may not be possible to create a copy of the unknown state of a quantum system. However, the system may be entangled with other quantum systems, effectively spreading the information in the system across several entangled objects.

[0027] This application relates to improved quantum error correction techniques for correcting errors in the state of a quantum system exhibiting one or more boson modes. In this context, an "error" refers to a change in the state of a quantum system, which may be caused, for example, by a boson loss, a boson gain, dephasing, time evolution, etc. of the system, that changes the state of the system such that the information stored in the system is changed.

[0028] As mentioned above, quantum multi-level systems such as qubits exhibit quantum states that decoherence in about 100 μs based on current experimental practice. It may therefore be beneficial to couple a multi-level system to another system that exhibits significantly longer decoherence times. As described below, boson modes are particularly desirable for coupling to a multi-level system. Through this coupling, the state of the multi-level system may instead be represented by a boson mode(s), thereby preserving the same information in a longer-lived state than would otherwise exist in the multi-level system alone.

[0029] Nevertheless, quantum information stored in bosonic modes still has a limited lifetime, and errors still occur in bosonic systems. It may therefore be desirable to manipulate the bosonic system when errors occur in its state to effectively correct these errors and thereby restore the previous state of the system. If a wide range of classes of errors could be corrected, it may be possible to maintain the state of a bosonic system indefinitely (or at least for a long time) by correcting any kind of error that may occur.

[0030] The fields of cavity quantum electrodynamics (cavity QED) and circuit QED represent one exemplary experimental approach for performing quantum error correction. In these approaches, one or more qubit systems are each coupled to a resonator cavity in such a way that allows mapping of quantum information contained in the qubit(s) to and / or from the resonator(s). The resonator(s) generally have a longer stable lifetime than the qubit(s). The quantum state can then be restored in the qubits by mapping the state back from the respective resonators to the qubits.

[0031] When a multi-level system such as a qubit is to be mapped onto the states of a bosonic system to which it is coupled, a particular 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."

[0032] As an example, the code may use the 0 boson number state of the resonator to represent the ground state of the qubit and the 1 boson number state of the resonator to represent the excited state of the qubit, i.e.:

number

[0033] The code usage is more general:

number

number

number

[0034] When an error occurs, the state of the system is transformed into a superposition of resulting states, referred to herein as the "error word", as follows:

number

number

[0035] However, one difficulty with the above approach is that the code may be limited by the lifetime of the nonlinear ancilla required for quantum control of the bosonic system. Typically, the bosonic system is controlled and errors in the bosonic system are corrected by manipulating an ancilla qubit that is coupled to the bosonic system. However, this may mean that if an error occurs in the ancilla qubit, error correction of the state of the bosonic system may no longer be possible.

[0036] The present inventors recognize and understand that a 4-legged Cat code can provide a fault-tolerant platform for performing quantum computational operations in a hardware-efficient quantum computing system. In particular, the present inventors have developed a universal set of operations for the 4-legged Cat code based on measurements of logical qubits and / or ancilla qubits. This universal gate set preserves fault tolerance against most possible first-order errors in logical qubits and ancilla qubits, including ancilla damping and phase relaxation.

[0037] We have developed a set of universal operations based on fault-tolerant parity operations for boson systems. In particular, we have extended the use of fault-tolerant parity measurements so that the Z, ZZ and ZZZ logical operators can be measured non-destructively and fault-tolerantly in a four-leg cat code. The implementation of these logical operators involves detuned beam splitter interactions, and the ancillary is in a superposition state to measure these operators. In some embodiments, the ZZ and ZZZ operators can also be measured when the ancillary is directly coupled to only one logical qubit of the multiple logical qubits.

[0038] We have further developed methods for preparing Z and X eigenstates, Bell states, and GHZ states in the four-leg Cat code using fault-tolerant parity measurements and the extensions described above. We have also developed methods for performing robust measurements in the Z, X, ZZ, and XX logic standards by combining beam splitter and cavity photon number measurements. For example, performing an X measurement uses interference of logic and coherent states using beam splitter interactions. Thus, photon number selective drive waveforms are applied to an ancilla qubit to determine whether one of the logic qubits (e.g., the cavity) is in the vacuum state. These measurements are fault-tolerant to all orders of transmon damping and dephasing errors in the sense that the overall measurement error can be exponentially suppressed by repeating the measurements and taking a majority vote on the results.

[0039] The inventors further recognize and understand that when combined with cavity displacement operations, this set of operators is sufficient for Clifford operations in a four-leg cat code while maintaining first-order fault tolerance against quantum errors. To make this set universal, the inventors have developed operations involving fault-tolerant SNAP gates to achieve any single-qubit Z rotation or, alternatively, the preparation of high-fidelity arbitrary states on the single-qubit Bloch sphere via a distillation scheme. This involves generating N imperfect copies of a target state and comparing the copies pairwise by performing non-destructive fault-tolerant SWAP tests between all possible pairs. Post-selection for passing all SWAP tests results in N copies of the state having higher fidelity to the target than the initial state.

[0040] The inventors further recognize and understand that single photon losses and jump-free backactions can be corrected in a four-leg Cat code via a teleportation scheme ("remote correction"). This scheme can be separated into two parts: generation of appropriate entangled Bell pairs and measurement in the Bell criterion. Thus, the inventors have developed techniques for generating Bell states and for performing Bell measurements on a four-leg Cat code. Such Bell states are then used to correct the jump-free backactions, which define the teleportation, while simultaneously correcting the single photon losses.

[0041] According to some aspects, the codes described herein can be used to configure the state of a bosonic system. A bosonic system can be a particularly desirable system for which the techniques described herein can be applied, which can exhibit equidistantly spaced coherent states as a single bosonic mode. For example, a resonator cavity is a simple harmonic oscillator with equidistant level spacing. Bosonic modes are also useful for quantum communication in that they can be stationary for quantum memory or for interacting with conventional qubits, or they can propagate ("fly") for quantum communication (e.g., they can be captured in and released from a resonator).

[0042] I. Exemplary Hardware Implementation 1 illustrates an exemplary system 100 suitable for implementing aspects of the present application. In system 100, quantum system 101 includes an ancilla qubit 110 that is coupled to a logical qubit 120 via dispersive coupling. That is, the ancilla qubit to logical qubit detuning is significantly larger (e.g., an order of magnitude larger) than the coupling strength between ancilla qubit 110 and logical qubit 120. Logical qubit 120 is also coupled to logical qubit 140 by beam splitter 130 (e.g., a programmable beam splitter). Energy source 150 may provide energy to one or all of ancilla qubit 110, logical qubit 120, beam splitter 130, and / or logical qubit 140 to perform operations on the system, such as adjusting a state in any of logical qubits 120 and / or 140, measuring one or more of logical qubits 120 and / or 140, applying a gate operation to one or more of logical qubits 120 and / or 140, applying an operation to or adjusting a state in ancilla qubit 110, detecting and / or correcting errors in ancilla qubit 110 and / or logical qubits 120 and / or 140, or combinations thereof.

[0043] According to some embodiments, logical qubit 120 and logical qubit 140 may be implemented as any suitable multi-mode boson system, which may include photonic systems such as one or more microwave cavities, although the techniques described herein are not limited to such systems. Logical qubit 120 and logical qubit 140 may be implemented as multi-mode boson systems that may include any combination of multiple modes of a single boson system and / or a single mode of multiple boson systems.

[0044] According to some embodiments, ancilla qubit 110 may include any suitable quantum system having three distinct states, such as, but not limited to, those based on superconducting Josephson junctions, such as, but not limited to, charged qubits (Cooper pair boxes), flux or phase qubits, transmon qubits, or combinations thereof. Ancilla qubit 110 may be coupled to logical qubit 120 via a dispersive coupling that couples the state of ancilla qubit 110 to the state of logical qubit 120. Logical qubit 120 may include any bosonic system supporting multiple bosonic modes, which may be implemented using any electromagnetic, mechanical, magnetic (e.g., quantized spin waves, also known as magnons), and / or other techniques, such as, but not limited to, any cavity resonator (e.g., microwave cavity). According to some embodiments, logical qubit 120 may include multiple transmission line resonators.

[0045] According to some embodiments, beam splitter 130 may be configured to provide a switchable beam splitter interaction between logical qubit 120 and one or more logical qubits 140. For example, each beam splitter 130 may be configured to provide a switchable beam splitter interaction between logical qubit 120 and one of logical qubits 140, the Hamiltonian of the form

number

[0046] System 100 also includes energy source 150, controller 160, and storage medium 170 (e.g., a computer-readable storage medium). In some embodiments, a library of pre-computed drive waveforms 172 may be stored on storage medium 170 and accessed by controller 160 to apply the waveforms to quantum system 101. For example, controller 160 may access drive waveforms 172 stored in storage medium 170 (e.g., in response to user input provided to the controller) and then control energy source 150 to apply one or more drive waveforms to each of ancilla qubit 110, logical qubit 120, beam splitter 130, and / or logical qubit 140.

[0047] As used herein, the application of such electromagnetic signals or pulses may also be referred to as "driving" the ancilla qubits and / or logical qubits. The coupling may utilize any technique or techniques for coupling the ancilla qubits and logical qubits, such as by coupling electric and / or magnetic fields generated by the ancilla qubits and logical qubits. According to some embodiments, the ancilla qubits (e.g., transmons) may be coupled to logical qubits and become mechanical resonators via piezoelectric coupling. According to some embodiments, the ancilla qubits may be coupled to logical qubits and become magnetic resonators by coupling the ancilla qubits (e.g., transmons) to photons, which in turn couple magnons via magnetostrictive coupling.

[0048] 2 shows an alternative exemplary system suitable for implementing aspects of the present application. In system 200, quantum system 201 includes an ancilla qubit 110 that is coupled to a logical qubit 140 via dispersive coupling. The logical qubit 140 is also coupled to another logical qubit 140 by a beam splitter 130. This beam splitter 130 can switch the beam splitter interaction on and off between any pair of logical qubits 140. Energy source 150 may provide energy to one or all of ancilla qubit 110, beam splitter 130, and / or logical qubit 140 to perform operations on the system, such as preparing a state in any one of logical qubits 140, measuring the state of one or more logical qubits 140, applying a gate operation to one or more logical qubits 140, applying an operation to ancilla qubit 110, detecting and correcting errors in ancilla qubit 110 and / or logical qubit 140, or combinations thereof.

[0049] II. Operation for 4-leg cat code Boson quantum computing encodes quantum information in the degrees of freedom of harmonic oscillators. By doing so, quantum error correction can be performed in a hardware-efficient manner; that is, quantum errors occurring in oscillators can be corrected without much additional physical hardware. One such encoding is the four-leg cat code, which is designed to correct single-photon loss errors in oscillators and is the dominant error channel in some quantum systems, such as quantum electrodynamic circuit systems.

[0050] To use this encoding as a quantum memory, one prepares the logic states in an appropriate codeword, detects and corrects single photon loss errors, and then reads out the logic information from the quantum system. To further use this encoding for quantum computer calculations, a set of universal gates must be further implemented.

[0051] Without quantum control of the harmonic oscillator, neither quantum memory nor computation may be possible. To perform quantum control of the harmonic oscillator using a classical external drive, a source of nonlinearity may be added to the system. For example, an ancilla qubit (e.g., a transmon qubit) may be added to the quantum system, and the ancilla qubit is dispersively coupled to the harmonic oscillator (e.g., a microwave cavity resonator). Unfortunately, the ancilla qubit may be an additional source of error that may propagate to the information stored in the harmonic oscillator.

[0052] Because these errors generated by ancilla qubits are quantum in nature, they can be described as jump operators. Although there is an infinite number of possible quantum errors, correcting the most likely errors that can occur in this cavity-transmon system in the time window during the error correction process significantly improves computational performance. Such errors include single photon losses in a harmonic oscillator, single decays of excitations in ancilla qubits and / or dephasing of states stored by ancilla qubits. This set of errors can be succinctly described as:

number

number

number

[0053] The quantum operations described herein are fault-tolerant to the errors mentioned above if the operations are designed such that if one of these errors occurs it does not result in a logical error on the qubit in the harmonic oscillator. This condition can be met either by being able to correct the error at a later time or if the error has a negligible effect on the logical information stored in the harmonic oscillator.

[0054] The inventors recognize and understand that reaching such a level of fault tolerance required for universal quantum computing using the four-leg cat code can be achieved in the measurement-based quantum computing (MBQC) paradigm. In circuit model quantum computing, a gate is applied to a qubit that remains fixed throughout the computation. In contrast, MBQC proceeds by preparing qubits in an entangled resource state, including a multi-body entangled state known as a "cluster state". The cluster state can then be used to perform a computation by measuring the qubits in a specific criteria. Rather than directly performing logic gates, quantum operations can be separated into quantum state preparation and destructive measurements; these operations are then used to realize quantum gates and quantum error correction.

[0055] The first quantum operation to enable fault-tolerant quantum computer computation in the four-leg Cat code is state preparation in a harmonic oscillator (e.g., logical qubit 120 or 140 described herein with respect to FIG. 1 or 2). FIG. 3A is a schematic diagram of an exemplary quantum circuit 300 for fault-tolerant preparation of |+> states in a qubit, according to some embodiments of the technology described herein.

[0056] In some embodiments, quantum circuit 300 describes operations applied to a single qubit in sequential order read from left to right. At the leftmost, the qubit begins in a vacuum state (|vac>). A displacement 302 (D(α)) may then be applied to displace the state of the qubit to a coherent state (e.g., α=2-3). After displacing the state of the qubit, repeated parity measurements 304 may be performed. Fault tolerance against ancillary errors is achieved by requiring that repeated parity measurements yield the same measurement result. If different results are obtained from each parity measurement 304, it is inferred that an error has occurred and the state may be discarded. By requiring that the two measurement results match, the state |α>±|-α> may be prepared with fault tolerance against a set of errors.

[0057] In some embodiments, the quantum circuit 300 may be implemented using an exemplary quantum information processing system 310 shown in FIG. The quantum information processing system 310 includes a logical qubit 312, shown as a microwave cavity resonator. The logical qubit 312 is dispersively coupled to an ancilla transmon qubit 314. A readout resonator 316 (e.g., a microwave strip resonator) is coupled to the ancilla transmon qubit 314 and configured to provide input to and / or read out information from the ancilla transmon qubit 314.

[0058] In some embodiments, the parity measurement 304 can be described as a sequence of quantum operations applied to an ancilla qubit and a logical qubit, as shown in the example of Figure 3C. In the quantum circuit of Figure 3C, the ancilla qubit |g> is connected to the logical qubit |ψ L >. The quantum operations include a first π / 2 rotation 304a of the ancilla qubit, a unitary operation 304b applied to the logical qubit, a second −π / 2 rotation 304c of the ancilla qubit, and a measurement 304d of the state of the ancilla qubit.

[0059] These quantum operations can be physically implemented by applying a series 320 of drive waveforms to an ancilla qubit, as shown in the example of Figure 3D. The series 320 includes a first sequence 322a including drive waveforms including ge π / 2 and ef π pulses, and a subsequent second sequence 322b including drive waveforms including ef π and ge π / 2 pulses. The sequences 322a and 322b are separated by a time delay T Π After sequence 322b is completed, readout 324 of the state of the ancilla qubit is performed.

[0060] Another quantum operation for performing state preparation in a 4-leg Cat code is shown in FIG. 4A. In the example of FIG. 4A, a quantum circuit 400 is configured to prepare a 4-leg Cat state |α>±|iα>+1α>±|-iα>, which serves as a logical 0 and 1 codeword for a 4-leg Cat code according to some embodiments. The quantum circuit 400 starts with a logical qubit in the vacuum state (|vac>). Then, a displacement 302 (D(α)) can be applied as described with respect to FIG. 3A. To fault-tolerantly prepare the 4-leg Cat state, a series of parity measurements 304 and logical Z measurements 406 can be applied in the order shown in FIG. 4A.

[0061] In some embodiments, the logical Z measurement 406 may be performed by measuring the 4-parity of a logical qubit. This measurement determines whether the logical qubit contains 0, 4, 8, etc. photons or 2, 6, 10, etc. photons. If the qubit contains 0, 4, 8, etc. photons, the measurement produces a result of +1, but if the qubit contains 2, 6, 10, etc. photons, the measurement produces a result of -1. If the qubit contains an odd number of photons, the measurement produces a random result. Fault tolerance is again achieved by requiring that a pair of parity measurements 304 and a pair of logical Z measurements 406 agree for a successful state preparation attempt.

[0062] In some embodiments, the logical Z measurement 406 can be described as a sequence of quantum operations applied to an ancilla qubit and a logical qubit, as shown in the example of FIG. 4B. The logical Z measurement 406 can be performed as a unitary operation 304b described with respect to FIG. 3B with half a waiting time (T 4Π =π / 2χ=T Π 4C is identical to the series of driving waveforms 320, the only change being that the waiting time between the first sequence 322a and the second sequence 322b is T 4Π =π / 2χ.

[0063] In addition to preparing states in the logical qubits, the states of the logical qubits must be measured as part of performing a quantum computer computation. Figure 5 is a schematic diagram of an exemplary quantum circuit 500 for performing fault-tolerant measurements in the Z-standard of a four-leg Cat code, according to some aspects of the technology described herein. The quantum circuit 500 is a circuit for performing fault-tolerant measurements in the Z-standard of a four-leg Cat code, according to some aspects of the technology described herein. L > measurement 502. This measurement 502 may be destructive in the sense that it dephases the state stored in the logical qubit if a decay error occurs during measurement 502. In such an example, the state stored in the logical qubit cannot then be used for further logical operations, but it may continue to be measured to improve the overall measurement fidelity by majority voting of repeated measurement results.

[0064] In some embodiments, measurement 502 in the Z basis of the four-leg Cat code may be physically performed by applying optimal control pulses to the ancilla qubit to excite the ancilla qubit if and only if the logical qubit contains photons for n=0, 3, 4, 7, 8.... Alternatively, the ancilla qubit may be driven by a linear combination of selective π-pulses at appropriate frequencies to perform measurement 502.

[0065] 6 is a schematic diagram of an example quantum circuit 600 for performing fault-tolerant measurements in the X-reference of a four-leg Cat code, according to some embodiments. To perform fault-tolerant measurements in the X-reference, quantum circuit 600 includes the use of ancillary logic qubits, shown on the bottom line of the circuit diagram. The ancillary logic qubits may start in a vacuum state (|vac>) and then be prepared in a coherent state by displacement 602 (D(α)) (e.g., as described with respect to displacement 302 in FIG. 3C). Measurements in the X-reference are performed by distributing the coherent state and the logic qubit |ψ L Beam splitter interference 604 between |α>±|-α> and |iα>±|-iα> states distinguishes between |α>±|-α> and |iα>±|-iα> states. Measurements 606 and 608 (performed, for example, using selective π pulses) determine whether exactly one of the logical qubits and ancillary qubits contains a 0 photon. If exactly one of the logical qubits and ancillary qubits contains a 0 photon, then if the input state is |iα>±|-iα>, then

number

[0066] 7 is a schematic diagram of an example quantum circuit 700 for performing a fault-tolerant measurement in the XX standard of the four-leg Cat code, according to some embodiments. A fault-tolerant measurement in the XX standard is very similar to the measurement 600 described with respect to FIG. 6. Instead of using a logical qubit and an ancillary qubit, the measurement of the quantum circuit 700 uses two logical qubits |ψ L1 > and |ψ L2If measurements 606 and 608 measure one of the logical qubits to contain zero photons, this indicates that the two-leg cats are both aligned along the same direction in phase space.

[0067] 8A is a schematic diagram of an exemplary quantum circuit 800 for performing a fault-tolerant measurement in the ZZ standard of a four-leg Cat code, according to some embodiments. In some embodiments, the quantum circuit 800 includes a measurement 802 of concatenated 4 parity of two logical qubits.

[0068] One way to measure the concatenated 4-parity of two logical qubits is to have an ancilla qubit coupled to a single cavity mode stored in the logical qubit. A single-mode 4-parity measurement sequence can then be performed without measuring any states. A SWAP operation can then be applied as described herein with respect to FIG. 19, followed by another single-mode 4-parity measurement. The ancilla qubit can then be measured in X-reference and another SWAP operation performed. However, this process faces the limitation that the beam splitter speed is typically smaller than χ, and χ needs to be cancelled out during the SWAP operation.

[0069] A faster sequence that avoids this problem is to combine the SWAP operation and the dispersive Hamiltonian into a single operation that performs a joint 4-parity measurement. To see how this works, first consider the joint 4-parity operator as a joint rotation by 90° in the cavity phase space:

number

number

number

number

number

[0070] These particular proportions may achieve the desired unitarity. This measurement may be made fault-tolerant to transmon errors in the same manner as the fault-tolerant parity measurement described with respect to Figs. 3A-3D. The use of the gf manifold of an ancilla qubit allows detection of transmon decay errors when |e> is measured using a χ-fit. This measurement also does not dephase the cavity even in the presence of single-transmon decay. Assuming that the parity measurement is used to track parity, and thus the 4-parity measurement is updated before the next parity jump occurs, photon loss may be correctable. For example, an ancilla qubit may be read out in the y basis when the cavity is odd by adding a 90° phase offset to the final π / 2 pulse.

[0071] Returning to Figure 8A, a concatenated 4-parity measurement 802 determines, for example, whether the total number of photons present in two logical qubits is n=0, 4, 8, ... or n=2, 6, 10, .... Measurement 802 is then followed by parity measurements 304 performed on each of the logical qubits, as described with respect to Figures 3A-3D. These parity measurements 304 are used to determine whether any photon losses have occurred in any of the logical qubits.

[0072] In some embodiments, the quantum circuit 800 may be implemented using an exemplary quantum information processing system 810 shown in FIG. 8B. The quantum information processing system 810 includes a first logical qubit 812a and a second logical qubit 812b. Both may be microwave cavity resonators, as shown in the example of FIG. 8B. The first and second logical qubits 812a, 812b are coupled to each other by a beam splitter 814. The first logical qubit 812a is dispersively coupled to an ancilla transmon qubit 816. A readout resonator 818 (e.g., a microwave strip resonator) is coupled to the ancilla transmon qubit 816 and configured to provide an input to the ancilla transmon qubit 906 and / or read out information from the ancilla transmon qubit 816.

[0073] In some embodiments, a measurement 802 of concatenated 4-parity of two logical qubits can be described as a sequence of quantum operations applied to an ancilla qubit and two logical qubits, as shown in the example of Figure 8C. In the quantum circuit of Figure 8C, the operations on the ancilla qubit |g> are applied to the logical qubit |ψ L1 > and |ψ L2 > is shown on the line below the state |ψ L1 The logical qubit that stores |ψ is a logical qubit that is dispersively coupled to an ancilla qubit, L1 > and |ψ L2> are coupled by a beam splitter as described for the example of Figure 8B. The quantum operations consist of a first π / 2 rotation 802a of the ancilla qubit, a second π / 2 rotation 802b of the logical qubit |ψ L1 > and |ψ L2 >, a second −π / 2 rotation 802c of the ancilla qubit, and a measurement 802d of the state of the ancilla qubit.

[0074] The quantum operation of Figure 8C may be physically implemented by applying a series 820 of drive waveforms to an ancilla qubit, as shown in the example of Figure 8D. The series 820 includes a first sequence 822a including drive waveforms including ge π / 2 and ef π pulses, and a subsequent second sequence 822b including drive waveforms including ef π and ge π / 2 pulses. The sequences 822a and 822b are time-delayed.

number

number

number

[0075] FIG. 9A is a schematic diagram of an exemplary quantum circuit 900 for performing fault-tolerant measurements in the ZZZ standard of a four-leg Cat code, according to some embodiments. Quantum circuit 900 can be considered an extension of quantum circuit 800, including a three-qubit measurement instead of a two-qubit measurement. To perform the ZZ measurement of quantum circuit 800, a switchable beam splitter interaction between a1 and a2 and a three-level ancilla qubit dispersively coupled to a1 was used. To extend this to the ZZZ measurement of quantum circuit 900, an additional switchable beam splitter coupling can be added between logical qubits a1 and a3, where a3 is the field operator of the third logical qubit. The target unitary during the π / 2 pulse is:

number

[0076] Performing two successive pairwise beam splitter interactions results in the following unitary:

number

[0077] From this equation, it can be observed that the first logical qubit has an additional conditional phase accumulated. When measured after performing these two beam splitter interactions, the operator Π1Z2Z3, where Π is the photon number parity of the first logical qubit. To cancel this, waiting a time T=π / (2χ) gives the following unitary:

number

number

[0078] Returning to FIG. 9A, in some embodiments, the quantum circuit 900 may be implemented using an exemplary quantum information processing system 910 shown in FIG. 9B. The quantum information processing system 910 includes a first logical qubit 912a, a second logical qubit 912b, and a third logical qubit 912c. All three qubits 912a, 912b, and / or 912c may be microwave cavities as shown in the example of FIG. 9B. The first and second logical qubits 912a, 912b are coupled to each other by a beam splitter 914a. The first and third logical qubits 912a, 912c are coupled to each other by another beam splitter 914b. The first logical qubit 912a is dispersively coupled to an ancillary transmon qubit 916. A readout resonator 918 (eg, a microwave strip resonator) is coupled to the ancilla transmon qubit 916 and configured to provide input to and / or read out information from the ancilla transmon qubit 916.

[0079] In some embodiments, a measurement 902 of three logical qubits can be described as a sequence of quantum operations applied to an ancilla qubit and three logical qubits, as shown in the example of Figure 9C. In the quantum circuit of Figure 9C, the operations on the ancilla qubit |g> are performed on the logical qubit |ψ L1 >, |ψ L2 > and |ψ L3 > is shown on the line below the state |ψ L1 A logical qubit that stores |ψ is a logical qubit that is dispersively coupled to an ancilla qubit, and the pair of logical qubits |ψ L1 > and |ψ L2 > and |ψ L1 > and |ψ L3> are each coupled by a beam splitter as described for the example of Figure 9B. The quantum operations are: a first π / 2 rotation 902a of the ancilla qubit; L1 > and |ψ L2 > beam splitter operation 902b applied to the logical qubit |ψ L1 > and |ψ L3 a second beam splitter operation 902c applied to the first logical qubit |ψ L1 > a unitary operation 902d applied to the ancilla qubit; a second −π / 2 rotation 902e of the ancilla qubit as well as a measurement 902f of the state of the ancilla qubit.

[0080] The quantum operation of Figure 9C may be physically implemented by applying a series of drive waveforms 920 to the ancilla qubit and beam splitter, as shown in the example of Figure 9D. The series 920 includes a first sequence 922a including drive waveforms including ge π / 2 and ef π pulses, and a subsequent second sequence 922b including drive waveforms including ef π and ge π / 2 pulses. The sequences 922a and 922b are separated by a time delay of 2T ZZ +T 4Π This time delay is 2T ZZ +T 4Π During the period, a drive waveform is applied to two beam splitters coupling a pair of logical qubits to define a detuned beam splitter interaction having a Hamiltonian of the form described above. 4Π can be used to correct for any rotation of the states stored in the accumulated logical qubits (e.g., −90°, −45° and −45° for the first, second and third logical qubits, respectively).

[0081] The ZZZ measurements of Figures 9A-9C can be used to perform CNOT gates when combined with the other quantum operations mentioned above, thus completing the Clifford gate set for the four-leg Cat code. This can be shown to be possible by using ZZZ measurements on the separable state |+++> to generate the entangled state |+++>+|--->. A similar entangled state |000>+|111> can be generated by measuring a pair of ZZ operators (e.g. Z1Z2 and Z2Z3) on the same initial state |+++>. Bell-based measurements on these states can deterministically perform CNOT gates, up to local Pauli corrections, forming a gate set that is known to be universal.

[0082] 10 is a flow chart describing a process 1000 for performing quantum operations according to some embodiments described herein. The process 1000 may be used to operate a quantum information processing system including, for example, a circuit quantum electrodynamics component. The quantum information processing system may include an ancilla qubit (e.g., a transmon qubit, a SNAILmon qubit, an oscillator, or another qubit) coupled to a first logical qubit (e.g., a microwave cavity). The first logical qubit may be coupled to a second logical qubit by a first beam splitter.

[0083] In some embodiments, process 1000 includes applying one or more drive waveforms to the ancilla qubit and / or the first beam splitter. The drive waveforms may be stored (e.g., locally or remotely) on one or more computer-readable storage media and accessed by the 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 the drive waveforms to the ancilla qubit and / or the first beam splitter.

[0084] In some embodiments, process 1000 may begin at act 1010, where a first drive waveform may be applied to the ancilla qubit. The first drive waveform may be:

number

[0085] In some embodiments, after act 1010, process 1000 may proceed to act 1020. In act 1020, a second drive waveform may be applied to the first beam splitter to define a detuned beam splitter interaction between the first logical qubit and the second logical qubit. The detuned beam splitter interaction may be

number

number

number

[0086] In some embodiments, after act 1020, process 1000 may proceed to act 1030, where a third drive waveform may be applied to the ancilla qubit. The third drive waveform may be:

number

[0087] In some embodiments, after act 1030, process 1000 may proceed to act 1040, where the state of the ancilla qubit may be read out. In some embodiments, the state of the ancilla qubit may be read out using a readout cavity or microwave strip resonator coupled to the ancilla qubit. To read out the state of the ancilla qubit, a measurement of the state of the ancilla qubit may be made. For example, a destructive measurement of the state of the ancilla qubit may be made. In some embodiments, this measurement may be made using a microwave irradiation detector that may distinguish between possible states of, for example, the readout cavity or microwave strip resonator. For example, the microwave irradiation detector may be a homodyne detector or a heterodyne detector in some embodiments.

[0088] III. Remote correction In standard quantum teleportation, an unknown state is "teleported" to a novel physical system. This teleportation can be performed using two steps. First, an entangled Bell pair can be generated. Second, a measurement of the unknown state and one-half of the Bell pair in the Bell basis is performed. Comparable to the well-known Pauli correction, which depends on the measurement result, the unknown state is deterministically teleported to the other half of the Bell pair after this measurement.

[0089] One problem that has long been an unsolved difficulty with the four-leg cat code is the so-called "jump-free" backaction that reduces the cat's "size", α, with time. If the correct Bell state can be generated, this jump-free backaction can be alleviated by teleporting the quantum information to a new logical encoding with a larger α. For example, if the first logical qubit initially starts with a cat size of α0, and the first logical qubit then decreases in size to a cat with an energy loss rate of κc If, after time t, the cat has a valid

number

[0090] By generating a Bell state between a qubit in a logical reference with α=α′ and a qubit in the same or another logical reference with α=α, a Bell state suitable for correcting the jump-free backaction can be generated. Quantum circuit 1100 of FIG. 11 illustrates the fault-tolerant preparation of Bell states for this purpose, according to some embodiments.

[0091] Quantum circuit 1100 begins with the preparation of two arbitrary states in two logical qubits. The first logical qubit can be transposed by a transition 1102 (D1(α)) and the second logical qubit can be transposed by a transition 1104 (D2(β)) to prepare two quantum states in the first and second logical qubits. In some embodiments, the first and second logical qubits can have initialized states that are in different logical references. In the example of FIG. 11, the first logical qubit is in a Cat code of “size” α and the second logical qubit is in a Cat code of size β. Preparing the two logical qubits in different logical references allows for jump-free backaction correction.

[0092] Two parity measurements 304 may then be performed as described with respect to Figures 3A-3D, one on each of the first and second logical qubits. The quantum circuit 1100 may then continue with two successive ZZ measurements as described with respect to Figures 8A-9C. Two further parity measurements 304 may then be performed on each of the first and second logical qubits. As described herein, to ensure fault-tolerant generation of Bell states, the first parity measurement 304 and the second parity measurement 304 must match for each of the first and second logical qubits. Furthermore, to ensure fault tolerance, both ZZ measurements 802 must also match.

[0093] The prepared Bell state 1100 is then transformed into a logical qubit |ψ by performing a measurement in the Bell standard, as shown in the quantum circuit of FIG. L > α The beam splitter interaction 1202 first couples the first qubit of the Bell state 1100, which is prepared with a cat code size of α, with the logical qubit |ψ L > α , and the logical qubit may be defined between |ψ|. Then, both the first qubit of Bell state 1100 and the logical qubit may be measured in the Z basis of the four-leg Cat code using measurement 1204. In some embodiments, measurement 1204 may be equivalent to measurement 502 described herein with respect to FIG. 5. Then, both the first qubit of Bell state 1100 and the logical qubit may be measured in the XX basis of the four-leg Cat code using measurement 1206, which may be equivalent to measurement 606 described herein with respect to FIG. 6. These measurements may then be taken of the originally logical qubit |ψ|. L > to a second qubit in Bell state 1100 with a Cat code of size β, correcting the jump-free backaction and preventing the accumulation of leakage errors over many quantum operations.

[0094] Since the beam splitter preserves total photon number parity (i.e., it preserves the number of photons), the ZZ information can be further extracted by measuring the local photon number parity mod(4) and adding the results to determine the ZZ information. After the beam splitter, all logical XX and ZZ information has been mapped into the nonlocal photon number space of the cavity, so the protocol is fault-tolerant. Ancillary qubit errors can further dephase logical qubits during the remote correction process, and at least two photon losses in either cavity are required to produce an incorrect measurement result.

[0095] One potentially useful subroutine specific to the cluster state model of quantum computing is the generation of Greenberger-Horne-Zeilinger (GHZ) entangled states such as |000>+|111> and |+++>+|--->. FIG. 13 is a schematic diagram of an exemplary quantum circuit 1300 for preparing a |000>+|111> GHZ cluster state, according to some embodiments described herein. The quantum circuit 1300 begins by applying a transposition 302 (D1(α), D2(α), D3(α)) to each logical qubit to prepare three arbitrary states in three logical qubits. A parity measurement 304 is then performed on each of the logical qubits. A first pair of ZZ measurements 802 is performed on the first and second qubits, and a second pair of ZZ measurements 802 is then performed on the second and third qubits. Finally, a parity measurement 304 is performed on each of the logical qubits. As before, the first and last parity measurements must match to provide fault tolerance, the first pair of ZZ measurements 802 must match, and the second pair of ZZ measurements 802 must match to prepare the |000>+|111>GHZ cluster state.

[0096] 14 is a schematic diagram of another example quantum circuit 1400 for preparing a |+++>+|--->GHZ cluster state according to some embodiments described herein. The quantum circuit 1400 is similar to the quantum circuit 1300 of FIG. 13, but instead of two pairs of ZZ measurements 802, a pair of ZZZ measurements 902 is performed between two sets of parity measurements 304. In this example, each of the ZZZ measurements 902 must match to preserve fault tolerance.

[0097] Combining GHZ state preparation with Bell state measurement allows the implementation of fault-tolerant CNOT gates between logical qubits. As with most teleported gate protocols, this can be divided into two steps: the generation of a suitable entangled state, followed by a Bell measurement between this entangled state and the logical qubit to simultaneously teleport the information to the remaining unmeasured qubits, and to execute the gate. This gate can be performed with several Pauli corrections that depend on the measurement outcome.

[0098] To prepare a CNOT gate, a first |χ〉 state can be prepared as shown in the example quantum circuit 1500 of FIG.

number

number

[0099] Once the |χ> state is prepared, it can be used to teleport a CNOT gate as shown in quantum circuit 1600 of FIG. 16 according to some embodiments. The CNOT gate can be teleported using Bell measurement 1200 between a pair of logical qubits 1 and 2 and a pair of logical qubits 5 and 6. The output of quantum circuit 1600 is stored as logical information on the unmeasured qubits 3 and 4, and the Bell measurement result indicates which Pauli correction should be applied to the output CNOT |Ψ2Ψ1> in any case. It should be understood that although there may be more efficient ways to compile quantum circuits using CNOT gates that reduce the number of operations and measurements, this explicit construction is useful to provide that the set of operations described herein is practically universal.

[0100] Combining state preparation and Bell state measurement also enables the implementation of a fault-tolerant Hadamard gate between two logical qubits. FIG. 17A illustrates the Hadamard state |Φ Had FIG. 17B is a schematic diagram of a quantum circuit 1700 for preparing

[0101] The precursor two-qubit entangled state is |Φ Had >. This state is also an eigenstate of the XZ operator, |0+>±|1->, |+i+i>±i|-ii> or H2|ψ 12 The quantum circuit 1700 can be written as:

number

[0102] One of the qubits is then destructively measured in the X-reference using measurement 600 that utilizes an additional ancilla qubit initialized in a different logical reference than the three logical qubits. Measurement 600 involves defining a beam splitter interaction 1708 between one of the logical qubits and the ancilla qubit and then destructively measuring the state of the logical qubit and the ancilla qubit using measurement 606. By destructively measuring this logical qubit in the X-reference, the two-qubit state of the other two logical qubits is projected to a state |+i+i>±i|-ii>, where the sign is determined by both the ZZZ measurement 902 and the result of the X-measurement 600.

[0103] |Φ Had After preparing the state |ψ, it can be used to teleport a single-qubit Hadamard gate to another logical qubit, as shown in quantum circuit 1800 of FIG. L > and a logical qubit with two qubits |Φ Had By performing the fault-tolerant Bell measurement 1200, the single-qubit Hadamard gate can measure the state of the two qubit |Φ HadAfter performing the fault-tolerant Bell measurement 1200, the two qubits |Φ Had The second qubit in the state is now in quantum state

number

[0104] The protocol described with respect to Figures 17A-18 utilizes a minimum of five logical qubits (e.g., five microwave cavities) with an ancillary qubit coupled to each of them. This implementation of the Hadamard gate is not particularly hardware efficient, but it is a novel implementation of the CNOT and R Z It should be appreciated that in combination with the θ operations, the above set of quantum operations is shown to be universal.

[0105] IV. State refinement using SWAP test Refining via SWAP test refers to a general method for symmetrizing a general qubit. The inventors recognize and understand that this method can be used to prepare states in boson qubits with high fidelity. In particular, non-destructive SWAP tests between pairs of states can be used to reduce errors in generating several copies of a target state using a procedure that is prone to error (e.g., that is noisy). SWAP test results can then be later selected to reduce errors in state generation.

[0106] This is a stand-alone procedure that can be used for general state preparation in boson modes to reduce the effect of stochastic errors in state preparation. As an alternative to using fault-tolerant SNAP gates as non-Clifford operations, it can be particularly useful to prepare |±i> and |T> states in the measurement system scheme described herein. Rather than directly implementing a fault-tolerant gate (e.g., a SNAP gate), fault-tolerant measurements can be used instead to refine noisy states created by some other means (e.g., optimal control pulses or state transfer from an ancillary to a logic qubit). The advantage of this method is that the noise channel can be complex and different for each input cavity state.

[0107] The SWAP test measurement can be made fault-tolerant to first-order errors, so that the initial state preparation errors can be much larger than the SWAP test errors. Under these conditions, the SWAP test can be used to refine the initial state and reduce the state preparation errors. The process begins by preparing N noisy copies of the desired quantum state to be initialized. For simplicity, let the probability of some error p err and the probability of no error occurring in state preparation (1-p err It can be estimated that when performing a SWAP test measurement between two of these cavities, there is a small probability p err / 2. Most of the time, the SWAP test measurement is successful, but with an error probability p err / 2. This direct tradeoff of probability of success for state fidelity is highly favorable.

[0108] By repeating the SWAP test over all the different pairings of the cavity, if the SWAP test measurement is successful, the error probability can be reduced until the limit set by the fidelity of the SWAP test measurement is reached. Since the SWAP test can be performed in a fault-tolerant manner, in principle this technique can be used to prepare cavity conditions with high fidelity.

[0109] To illustrate this method, we describe the construction of a single fault-tolerant SWAP test measurement from universal operations. FIG. 19 is a schematic diagram of an example quantum circuit 1900 for fault-tolerant implementation of a SWAP test between a first and a second qubit, according to some embodiments. A SWAP test in this context is a non-destructive measurement of the SWAP operator between two logical qubits. Since symmetric and asymmetric superpositions are ±1 eigenstates of the SWAP operator, if |Ψ1〉 and |Ψ2〉 are initial input states, then the SWAP test can be performed by dividing these states into

number

[0110] To perform this measurement, a 50-50 beam splitter interaction 1900a is first defined between two logical qubits. The photon number parity of one of the modes is then measured in the “beam splitter” frame using a parity measurement 304. Typically, a parity measurement on a single logical qubit is performed using the parity operator

number

number

number

[0111] At the surface value, the result of the SWAP test is fairly simple to interpret. If the result obtained is +1 (i.e. the ancilla qubit is in the |g〉 state), then it is more likely that the two input states |ψ1〉 and |ψ2〉 are identical and therefore free of errors. By choosing this result later, the probability that either state has an error is reduced accordingly.

[0112] The probability of getting the result ±1 is 1±|<Ψ1|Ψ2>| 2 / 2. If one of the states suffers an error in the initial preparation, then it is highly likely that |<ψ1|ψ2>|=0 for a wide range of possible errors. Furthermore, obtaining a result of +1 does not guarantee that no error occurred. If |<ψ1|ψ2>|=0, then a result of +1 can still be obtained with a probability of 0.5. Because of this, errors in both cavity states are reduced by half when passing the SWAP test, but are not completely eliminated.

[0113] This can be more accurately represented by density matrix formalism. The initial noisy cavity state is:

number

[0114] The initial two-cavity state is

number

number

[0115] 20 is a schematic diagram of an example quantum circuit 2000 configured to reduce errors present in a quantum state prepared in four qubits, according to some embodiments of the technology described herein. The quantum circuit 2000 includes several SWAP tests 1900 and SWAP operations 2002 between pairs of cavities. In the example of FIG. 20, the process init During execution of the quantum circuit 2000, the SWAP test 1900 detects errors in each state by p err ρ can be performed among all six permutations of pairs of logical qubits to reduce the init With additional copies of , this error rate could be further reduced at the expense of adding more SWAP tests and SWAP operations. This protocol can be experimentally run using the same hardware as the ZZ measurement, as described herein with respect to Figures 8A-8D.

[0116] V. The Kerr effect and the correction of χ' In some quantum information processing schemes, it may be desirable to account for additional effects that may introduce perturbations to the quantum system. For example, the Kerr effect and the χ' effect may result in fluctuating effects on the ZZ and / or ZZZ measurements described herein, such that they are not robust. These effects are particularly pronounced in systems that utilize many photons (e.g., 10 or more photons) because the frequency of the measured transitions between the states of the ancilla qubit depends on the number of photons stored in the logical qubit. For example, the χ' effect is quadratically proportional to the number of photons stored in the logical qubit, and the χ' effect is more difficult to distinguish and correct for higher photon numbers. These effects are particularly important to account for in MBQC, where many more photons are used to perform computational processes.

[0117] The Kerr and χ' effects can be expressed by the following two-qubit Hamiltonian:

number

[0118] 21 is a schematic diagram of a Bloch sphere illustrating the effects of the Kerr effect and χ' on a quantum state, according to some embodiments. These two effects introduce perturbations 2102 and 2104 around the periphery of the Bloch sphere that can reduce the robustness of the ZZ and / or ZZZ measurements described herein and increase the probability of decoherence between quantum circuits.

[0119] To counter these effects, the quantum states stored in the logical qubits can be prepared using alternative processes, and the quantum operations can be changed as well. In some embodiments, the cat state can be prepared by first transitioning the state of the logical qubit from the vacuum state |vac> to the state |α>. The |α> state can then be transitioned to |0> using a drive waveform that includes a selective gf π-pulse comb. LThe selective gf π-pulse comb can be a π-pulse containing multiple frequencies corresponding to frequencies (0χ, 4χ, 8χ, 12χ, ...). The use of these selective frequencies in the gf π-pulse comb addresses the effect of χ', and varying the phase of the constituent π-pulses addresses the Kerr effect perturbations, as these provide second-order corrections to the equally spaced energy levels of the logical qubits. An example of a selective gf π-pulse comb is shown in Figure 22A, and the corresponding Fourier spectrum is shown in Figure 22B.

[0120] In some embodiments, the measurements can also be tailored to counter the effects of χ' and the Kerr effect. For example, XX and ZZ information can be extracted simultaneously to perform Bell measurements using a three-level ancilla qubit (e.g., a three-level transmon qubit). To extract this information, three measurements can be performed. In some embodiments, these measurements can be performed simultaneously. First, information related to the |f> state can be measured using selective Raman transitions. Second, information related to the |e> state can be measured by driving the ancilla qubit with a drive waveform including π pulses including selective frequency combs with frequencies of (3χ, 4χ, 7χ, 8χ, ...). Third, information related to the |g> state can be measured by driving the ancilla qubit with a drive waveform including π pulses including selective frequency combs with frequencies of (1χ, 2χ, 5χ, 6χ, ...). An example of such a drive waveform including both frequency combs to reveal |e> and |g> states is shown in FIG. 23A. The corresponding Fourier transform is shown in FIG. 23B.

[0121] 24 is a flow chart describing another process 2400 for performing quantum operations according to some aspects of the technology described herein. The process 2400 may be used to operate a quantum information processing system including, for example, a circuit quantum electrodynamics component. The quantum information processing system may include an ancillary qubit (e.g., a transmon qubit, a SNAILmon qubit, an oscillator, or another qubit) coupled to a first logical qubit (e.g., a microwave cavity resonator).

[0122] In some embodiments, process 2400 may begin by act 2410, where a first drive waveform is generated and applied to an ancilla qubit. The drive waveforms described with respect to process 2400 may be stored (e.g., locally or remotely) in one or more computer-readable storage media and accessed by a controller. To apply the drive waveform, the controller may cause an energy source (e.g., a microwave source) to generate the drive waveform and communicate the drive waveform to the ancilla qubit and / or other components of the quantum information processing system.

[0123] In some embodiments, the first drive waveform includes a first comb of π pulses having selective frequencies corresponding to a first selection of even and odd cavity resonant frequencies of the first logical qubit, for example, the first comb of π pulses can have selective frequencies corresponding to frequencies of (3χ, 4χ, 7χ, 8χ, ...).

[0124] In some embodiments, the method optionally includes performing act 2420 prior to reading out the state of the ancilla qubit. Act 2420 may include generating a second drive waveform and applying it to the ancilla qubit. The second drive waveform may include a second comb of π pulses having selective frequencies corresponding to a second selection of even and odd cavity resonant frequencies of the first logical qubit. In some embodiments, the second comb of π pulses may have selective frequencies corresponding to selective frequencies of (1χ, 2χ, 5χ, 6χ...).

[0125] In some embodiments, after act 2410 or 2420, process 2400 may proceed to act 2440, where the state of the ancilla qubit may be read out. In some embodiments, the state of the ancilla qubit may be read out using a readout cavity or microwave strip resonator coupled to the ancilla qubit. To read out the state of the ancilla qubit, a measurement of the state of the ancilla qubit may be made. For example, a destructive measurement of the state of the ancilla qubit may be made. In some embodiments, this measurement may be made using a microwave radiation detector that may distinguish between possible states of, for example, the readout cavity or microwave strip resonator. For example, in some embodiments, the microwave radiation detector may be a homodyne detector or a heterodyne detector.

[0126] In some embodiments, performing the quantum operation includes measuring a Bell state between the first logical qubit and the second logical qubit. In such embodiments, the quantum electrodynamics system further includes a second logical qubit coupled to the first logical qubit by a first beam splitter. For example, the first logical qubit and the second logical qubit can each be a microwave cavity coupled to the first beam splitter. The method can include applying a third drive waveform to the first beam splitter to define a detuned beam splitter interaction between the first logical qubit and the second logical qubit before reading out the state of the ancilla qubit. Thereafter, process 2400 can proceed with act 2440 as described above.

[0127] In some embodiments, process 2400 further includes generating a first four-qubit cluster state. The four-qubit cluster state may be generated, at least in part, by applying a fourth drive waveform to a second beam splitter coupling the first logical qubit and the third logical qubit to define a beam splitter interaction between the first logical qubit and the third logical qubit. The four-qubit cluster state may further be generated by applying a fifth drive waveform to a third beam splitter coupling the second logical qubit to the fourth logical qubit. In this manner, the quantum states stored in the four logical qubits may be entangled to generate a four-qubit cluster state.

[0128] In some embodiments, process 2400 further includes generating a multi-qubit cluster state. The multi-qubit cluster state may be, for example, an XZZX cluster state described herein, or it may be 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 drive waveform to a fourth beam splitter that couples a first logical qubit of the first four-qubit cluster state and a first logical qubit of the second four-qubit cluster state.

[0129] VI. Cluster state preparation The inventors recognize and understand that the quantum operations described above can be used to generate cluster states suitable for MBQC. Once generated, the cluster states can be used to perform computations by measuring qubits in a particular manner. Alternatively or additionally, the cluster states are useful for quantum communication and networking.

[0130] Figure 25A is a schematic diagram of an example quantum circuit 2500 configured to prepare a Bell state in two qubits, according to some embodiments. Figure 25B is a schematic diagram of a two-qubit ZZ Bell state 2510 that can be prepared using the quantum circuit 2500, according to some embodiments. The illustration in Figure 25B includes two qubits 2512 prepared in a first logical reference represented by a closed circle. The line connecting the two qubits 2512 represents coupling due to entanglement.

[0131] The quantum circuit of 2500 starts with two logical qubits prepared in the |α> and |iα> states, respectively. The two logical qubits are coupled by a beam splitter, and the quantum circuit 2500 includes the creation of a beam splitter interaction 2504 between the two logical qubits. Before and after the beam splitter interaction 2504, parity measurements 304 are used to ensure fault tolerance. The Bell state |Φ Bell The generation of > is successful if Π1+Π2=Π3+Π4.

[0132] An example of a four-qubit cluster state can be generated by coupling beam splitter interactions as shown in Figure 26A. Figure 26B is a schematic diagram of a four-qubit cluster state 2610 that can be generated using quantum circuit 2600. The quantum circuit 2600 of Figure 26A

number

[0133] Another building block of cluster states for MBQC is a two-qubit cluster, consisting of two qubits, each prepared in a different logical reference by teleporting a Hadamard state. Figure 27A is a schematic diagram of a quantum circuit 2700 configured to generate the two-qubit entangled state 2710 shown in Figure 27B, according to some embodiments. The two-qubit entangled state 2710 includes a first qubit 2512 prepared in a first logical reference (e.g., X) and a second qubit 2714 prepared in a second, different logical reference (e.g., Z).

[0134] In some embodiments, the quantum circuit 2700 comprises:

number

number

[0135] 28A is a schematic diagram describing a fusion process that may be used to generate another four-qubit cluster state according to some embodiments of the technology described herein. The process may begin at stage 2800 with four individual cluster states including three two-qubit states and one four-qubit state. Bell measurement 2802, which may be any suitable Bell measurement described herein, may be used to "fuse" the qubits of each of these smaller resource states to produce a four-qubit cluster state 2810.

[0136] Quantum operations similar to those described with respect to Figures 25A-28B can be further coupled to generate larger cluster states that are useful for MBQC. Such cluster states can include, for example, XZZX cluster states described herein or alternatively or additionally RHG cluster states. Figure 28B is a schematic diagram illustrating the formation of XZZX cluster states according to some embodiments of the technology described herein.

[0137] As shown in the example of Figure 28B, 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 generate a final cluster state used for MBQC or other applications. As shown in Figure 28B, in some embodiments, the larger cluster state can be XZZX cluster state 2830. Further aspects of the XZZX cluster state are described in "Tailored cluster states with high threshold under biased noise" by J. Claes, J. Eli Bourassa, and S. Puri, submitted to ArXiv on January 25, 2022, and located at arXiv:2201.10566, which is incorporated by reference herein in its entirety.

[0138] An exemplary implementation of a classical computer system 2900 that may be used in connection with any of the embodiments of the disclosure provided herein is shown in FIG. 29. In some embodiments, any one of the processes described herein may be executed on and / or using the computer system 2900. The computer system 2900 may include one or more articles of manufacture including one or more processors 2910 and 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 and reading of data to and from the memory 2920 and the non-volatile storage device 2930 in any suitable manner. To perform any of the functions described herein, the processor 2910 may execute one or more processor-executable instructions stored in one or more non-transitory computer-readable storage media (e.g., memory 2920) that may act as a non-transitory computer-readable storage medium that stores processor-executable instructions for execution by the processor 2910.

[0139] Although several aspects and embodiments of the technology described in this disclosure are thus described, it is understood that various changes, modifications, and improvements will be readily implemented by those skilled in the art. Such changes, modifications, and improvements are intended to be within the spirit and scope of the technology described herein. For example, one skilled in the art will readily envision various other means and / or structures for performing the functions and / or obtaining the results and / or one or more of the advantages described herein, and each of such changes and / or modifications is considered to be within the scope of the embodiments described herein. One skilled in the art will recognize or be able to ascertain using no more than routine experimentation many equivalents to the specific embodiments described herein. Thus, it is understood that the foregoing embodiments are presented by way of example only, and that within the scope of the appended claims and their equivalents, the embodiments of the invention may be practiced other than as specifically described. Also, any combination of two or more features, systems, articles, materials, kits, and / or methods described herein is included within the scope of the present disclosure, if such features, systems, articles, materials, kits, and / or methods are not mutually inconsistent.

[0140] The above-described aspects may be implemented in any of many ways. One or more aspects and embodiments of the present disclosure, including the performance of a process or method, may utilize program instructions executable by a device (e.g., a computer, a processor, or other device) to perform or control the performance of the process or method. In this aspect, the various inventive concepts may be embodied as a computer-readable storage medium (or multiple computer-readable storage media) (e.g., a computer memory, one or more floppy disks, compact disks, optical disks, magnetic tapes, flash memory, circuitry within a field programmable gate array or other semiconductor device, or other tangible computer storage medium) encoded with one or more programs that, when executed on one or more computers or other processors, perform a method for performing one or more of the various aspects described above. The computer-readable medium(s) may be transportable such that the program(s) stored thereon may be loaded onto one or more different computers or other processors to perform various of the above-described aspects. In some embodiments, the computer-readable medium may be a tangible (e.g., non-transitory) computer-readable medium. In some embodiments, the computer-readable medium may include a persistent memory.

[0141] The term "program" or "software" is used herein in a general sense to refer to any type of computer code or set of computer executable instructions that can be used to program a computer or other processor to perform the various aspects described above.Furthermore, according to one aspect, it should be understood that one or more computer programs that, when executed, perform the methods of the present disclosure do not have to reside on a single computer or processor, but may be distributed in a modular manner among several different computers or processors to perform various aspects of the present disclosure.

[0142] Computer-executable instructions may exist in many forms, such as program modules, executed by one or more computers or other devices. Generally, program modules include routines, programs, objects, components, data structures, etc. that perform particular tasks or implement particular abstract data types. Typically the functionality of the program modules may be combined or distributed as desired in various embodiments.

[0143] When implemented in software, the software code may be executed on any suitable processor or collection of processors, whether provided on a single computer or distributed among multiple computers.

[0144] Further, it should be understood that a computer may be embodied in any of a number of forms, such as, by way of non-limiting examples, a rack-mounted computer, a desktop computer, a laptop computer, or a tablet computer. Additionally, a computer may be encompassed by devices not generally considered computers, but having suitable processing capabilities, such as a personal digital assistant (PDA), a smartphone, or any other suitable portable or fixed electronic device.

[0145] A computer may also have one or more input and output devices. These devices may be used, among other things, to present a user interface. Examples of output devices that may 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 audible presentation of output. Examples of input devices that may be used for a user interface include a keyboard and pointing devices, such as a mouse, touchpad, and digitizing tablet. As another example, a computer may receive input information via speech recognition or in other audible form.

[0146] Such computers may be interconnected by one or more networks of any suitable form, such as local area networks or wide area networks, e.g., enterprise networks and intelligent networks (IN) or the Internet, etc. Such networks may be based on any suitable technology and may operate according to any suitable protocol, and may include wireless networks, wired networks or fiber optic networks.

[0147] Also, as described, some aspects may be embodied as one or more methods. The acts performed as part of a method may be ordered in any suitable manner. Thus, even if shown as sequential acts in an example embodiment, the acts may be performed in an order different from that shown, and embodiments may be constructed that may include performing some acts simultaneously.

[0148] All definitions defined and used herein should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and / or ordinary meanings of the defined terms.

[0149] The indefinite articles "a" and "an," as used in the specification and claims, unless clearly indicated to the contrary, should be understood to mean "at least one."

[0150] The phrase "and / or" when used in the specification and claims should be understood to mean "either or both" of the elements so connected, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with "and / or" should be interpreted in the same manner, i.e., "one or more" of the elements so connected. Other elements, whether related or unrelated to those elements specifically identified, may optionally be present other than the elements specifically identified by the "and / or" clause. Thus, as a non-limiting example, a reference to "A and / or B", when used in conjunction with open-ended language such as "comprising", may refer in one embodiment to A only (optionally including elements other than B); in another embodiment to B only (optionally including elements other than A); in yet another embodiment to both A and B (optionally including other elements); and so forth.

[0151] As used in the specification and claims, the phrase "at least one" should be understood, in reference to a list of one or more elements, to mean at least one element selected from any one or more of the elements in the list of elements, without necessarily including at least one of each and every element specifically listed in the list of elements, but without excluding any combination of elements in the list of elements. This definition also allows for the optional presence of elements other than those specifically identified in the list of elements to which the phrase "at least one" refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, "at least one of A and B" (or, equivalently, "at least one of A or B) or, equivalently, "at least one of A and / or B") can refer, in one embodiment, to at least one that optionally includes more than one A in the absence of B (and optionally including elements other than B); in another embodiment, to at least one that optionally includes more than one B in the absence of A (and optionally including elements other than A); in yet another embodiment, to at least one that optionally includes more than one A, and at least one that optionally includes more than one B (and optionally including other elements); etc.

[0152] In the claims and the foregoing specification, all transitional phrases, such as "comprising," "including," "carrying," "having," "containing," "involving," "holding," "composed of," and the like, are understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases "consisting of" and "consisting essentially of" are intended to be closed or semi-closed transitional phrases, respectively.

[0153] The terms "approximately" and "about" can be used to mean, in some embodiments, within ±20% of a target value, in some embodiments, within ±10% of a target value, in some embodiments, within ±5% of a target value, and in some embodiments, within ±2% of a target value. The terms "approximately" and "about" can include the target value.

Claims

1. 1. A method of operating a circuit quantum electrodynamic system including an ancilla qubit dispersively coupled to a first logical qubit, the method comprising: At least in part: generating and applying a first drive waveform to the ancilla qubit, where the first drive 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 logical qubit; and Reading out the state of an ancilla qubit performing a quantum operation by

2. 13. The method of claim 1, further comprising generating and applying a second drive waveform to the ancilla qubit prior to reading out the state of the ancilla qubit, the second drive waveform comprising a second comb of π pulses having selective frequencies corresponding to a second selection of even and odd cavity resonant frequencies of the first logical qubit.

3. the first selection includes selective frequencies 3χ, 4χ, 7χ and 8χ; The second selection includes selective frequencies 1χ, 2χ, 5χ and 6χ; The method of claim 2.

4. 2. The method of claim 1, wherein the circuit quantum electrodynamic system further includes a second logical qubit coupled to the first logical qubit by the first beam splitter, and the method further includes applying a third drive waveform to the first beam splitter to define a detuned beam splitter interaction between the first logical qubit and the second logical qubit prior to reading out the state of the ancilla qubit.

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

6. 5. The method of claim 4, wherein defining a detuned beam splitter interaction between the first logical qubit and the second logical qubit comprises defining a detuned beam splitter interaction between the first resonant cavity and the second resonant cavity.

7. The method of claim 1 , wherein generating and applying a first drive waveform comprises generating and applying a microwave waveform.

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

9. At least in part: applying a fourth drive waveform to a second beam splitter coupling the first logical qubit and the third logical qubit; and applying a fifth drive waveform to a third beam splitter that couples the second logical qubit to the fourth logical qubit; 5. The method of claim 4, further comprising generating the first four-qubit cluster state by:

10. At least in part: applying a sixth drive waveform to a fourth beam splitter coupling the first logical qubit in the first four-qubit cluster state and the first logical qubit in the second four-qubit cluster state. The method of claim 9 , further comprising generating the multi-qubit cluster state by:

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

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

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

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

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

16. 16. The quantum information processing system of claim 15, wherein the at least one controller configured to perform quantum operations comprises at least a controller configured to generate Bell states between the first logical qubit and the second logical qubit.

17. 15. The quantum information processing system of claim 14, wherein the first logical qubit and the second logical qubit comprise a first resonant cavity and a second resonant cavity.

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

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

20. 1. A method of operating a circuit quantum electrodynamic system including an ancilla qubit dispersively coupled to a first logical qubit and a second logical qubit coupled to the first logical qubit by a first beam splitter, comprising: applying a first drive waveform to the ancilla qubit, where the first drive waveform comprises a π / 2 pulse; applying a second drive waveform to the first beam splitter to define a detuned beam splitter interaction between the first logical qubit and the second logical qubit; applying a third drive waveform to the ancilla qubit, where the third drive waveform includes a π / 2 pulse; and Reading out the state of an ancilla qubit A method comprising:

21. 11. A method of claim 1, wherein the circuit quantum electrodynamic system further comprises a third logical qubit coupled to the first logical qubit by a second beam splitter, the method comprising:

21. The method of claim 20, further comprising applying a fourth drive waveform to the second beam splitter after applying the second drive waveform to define a detuned beam splitter interaction between the first logical qubit and the third logical qubit.

22. 1. A method of operating a circuit quantum electrodynamic system including 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 being coupled to the second logical qubit by a first beam splitter, the method comprising: applying a first drive waveform to a first beam splitter to define an on-resonance beam splitter interaction between the first logical qubit and the second logical qubit; and applying a second driving waveform to the first ancilla qubit to measure the state of the first logical qubit; and applying a third driving waveform to the second ancilla qubit to measure the state of the second logical qubit. determining whether at least one of the first and second logical qubits is in a vacuum state by A method comprising:

23. 1. A method of operating a circuit quantum electrodynamic system including 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, wherein the first logical qubit and the second logical qubit are coupled by a first beam splitter, and the second logical qubit and the third logical qubit are coupled by a second beam splitter, the method comprising: Preparing an arbitrary logical state in a 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; Teleport to: introducing interference between the first logical qubit and the second logical qubit using a first beam splitter; and performing at least one measurement of the states of the first logical qubit and the second logical qubit using the first ancilla qubit and the second ancilla qubit after using the first beam splitter; A method comprising:

24. To prepare the Bell state: 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 concatenated parity measurements on the second logical qubit and the third logical qubit.

24. The method of claim 23, comprising:

25. Ancyla Cubitt; and a first logical qubit dispersively coupled to the ancilla qubit; and A second logical qubit coupled to the first logical qubit by a beam splitter. Multiple logical qubits including A circuit quantum electrodynamics system comprising:

26. 26. The circuit quantum electrodynamic system of claim 25, wherein the ancilla qubit comprises a transmon qubit.

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

28. 30. The circuit quantum electrodynamic system of claim 27, wherein a logical qubit of the plurality of logical qubits comprises a boson mode.

29. The circuit quantum electrodynamics system of claim 6; and preparing an arbitrary logical state in a first logical qubit; preparing a Bell state between the second logical qubit and the third logical qubit; at least one controller configured to perform error correction on any coherent state by teleporting the any logical state from the first logical qubit to the third logical qubit; A system comprising: Teleport to: introducing interference between the logical qubit and a second logical qubit using at least one beam splitter; and performing at least one measurement of the states of the first logical qubit and the second logical qubit using the first ancilla qubit and the second ancilla qubit after using the at least one beam splitter; Including, the system.

Images