Fault-tolerant quantum cat state preparation

Abstract

The quantum computing system is adapted to prepare a cat state in a quantum circuit having a fault tolerance t and a circuit depth of 4+4t or less by performing a sequence of operations, the sequence of operations including performing a joint parity measurement sequence on each pair of neighboring qubits of a set of entangled qubits to form an initial cat state, repeating the measurement sequence for at least t rounds, and disentangling a first set of alternating qubits from the initial cat state, wherein the prepared cat state is formed by a remaining second set of alternating qubits, the second set of alternating qubits being entangled with the first set of alternating qubits along one-dimensional connectivity lines, and the sequence of operations is sufficient to ensure that the prepared cat state has no more than t faults.

Description

[Background technology]

[0001] background

[0001] Schrödinger cat states, or simply "cat states", are formed by highly entangled quantum states of multiple qubits. Cat states are used in several types of quantum operations, including performing logic operations (e.g., both within a logic block and across different logic blocks), quantum error correction, and teleportation of quantum states from one bit to another, to provide fault-tolerant operations. For example, cat states can be used to perform Pauli operator measurements of two or more data qubits. Such a measurement entails (1) entangling a set of ancillary qubits through a sequence of operations to produce a cat state for those ancillary qubits, (2) entangling those qubits in the prepared cat state with the data qubit to be measured, and (3) performing a single-qubit measurement on each of the ancillary qubits to extract the data qubit measurement given by the collective parity of the bit on which the measurement is made. Summary of the Invention

[0002] overview

[0002] According to one implementation, a quantum circuit with fault tolerance t and circuit depth of 4+4t or less executes a cat state by performing a sequence of operations including: performing a joint parity measurement sequence on each pair of neighboring qubits of a set of qubits entangled to form an initial cat state; repeating the measurement sequence for at least t rounds; and disentangling a first set of alternating qubits from the initial cat state to form a prepared cat state with a remaining second set of alternating qubits, the second set of alternating qubits being entangled with the first set of alternating qubits along a one-dimensional connectivity line. The prepared cat state is guaranteed to contain no more than t faults with a predefined probability. Other implementations are also described and described herein. [Brief description of the drawings]

[0003] BRIEF DESCRIPTION OF THE DRAWINGS [Figure 1]

[0003] A quantum computing system including a small-depth fault-tolerant cat state preparation circuit is shown. [Figure 2A]

[0004] 1 illustrates a qubit lattice of a quantum computing device. [Figure 2B]

[0005] 2B shows an example of a fault-tolerant cat preparation circuit that prepares a cat state that can be used to measure the target data qubit of FIG. 2A. [Diagram 3]

[0006] 1 shows another example of a small depth fault tolerant cat state preparation circuit. [Figure 4A]

[0007] We show an example of a cat state preparation circuit that is t-fault tolerant and has a depth of 4+2t. [Figure 4B]

[0008] 1 shows another example of a cat state preparation circuit that is t fault tolerant and has a depth of 4+2t. [Figure 4C]

[0009] 1 shows yet another example of a cat state preparation circuit that is t fault tolerant and has a depth of 4+2t. [Figure 5A]

[0010] 1 shows an example of a rejection-free fault-tolerant cat-state preparation circuit that is t-fault tolerant and has a depth of 4+4t. [Figure 5B]

[0011] 1 shows another example of a rejection-free fault-tolerant cat-state preparation circuit that is t-fault tolerant and has a depth of 4+4t. [Figure 5C]

[0012] 1 shows yet another example of a rejection-free fault-tolerant cat-state preparation circuit that is t-fault tolerant and has a depth of 4+4t. [Figure 6A]

[0013] 4A-4C show an example of a cat state preparation circuit that is t fault tolerant and has a depth of 4+2t, with a reduced size compared to FIGS. [Figure 6B]

[0014] 4A-4C , depicting another example of a cat state preparation circuit that is t fault tolerant and has a depth of 4+2t. [Figure 6C]

[0015] 4A-4C , depicting yet another example of a cat state preparation circuit that is t fault tolerant and has a depth of 4+2t. [Figure 7]

[0016] We present examples of operations for preparing cat states using postselection techniques in fault-tolerant t and 4+2t deep circuits. [Figure 8]

[0017] 8 illustrates an example of operations 800 for preparing a no-rejection cat state in a fault-tolerant t and 4+2t deep circuit. [Figure 9]

[0018] 1 illustrates an exemplary computing environment suitable for implementing aspects of the disclosed technology. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0004] Detailed Description

[0019] Preparing the Schrödinger cat state entails performing a sequence of measurements to entangle a selected set of qubits. This process can be followed by a postselection stage in which a sequence of measurements is performed to verify that the cat state is prepared to a certain degree of sufficiency (e.g., the cat state contains a number of faults up to a predefined fault tolerance of the corresponding measurement circuit).

[0005]

[0020] Existing Schrödinger cat state post-selection (test) methods measure the parity of random pairs of qubits in the cat state. Ensuring that the cat state is prepared accurately enough entails many measurements and a high degree of qubit connectivity (e.g., connections between any and all auxiliary qubits used in the post-selection test and all qubits of the cat state). This high degree of connectivity is difficult to obtain with currently proposed quantum platforms, including, for example, in a lattice of qubits, using superconducting qubits, using Majorana qubits, or using logical qubits encoded with surface codes, color codes, or small code blocks. Moreover, this method for testing random pairs of qubits in the cat state entails a significant number of measurement rounds, each of which consumes a significant amount of processing time. For example, each measurement round can take about 100 ns for superconducting qubits, or 100 microseconds for ion traps.

[0006]

[0021] The techniques disclosed herein provide a circuit for cat state preparation and a test method for testing the prepared cat state that can be implemented with low processing time overhead (e.g., fewer measurement rounds) and reduced hardware requirements (e.g., lower degree of connectivity between qubits) compared to the existing test methods described above. According to one implementation, the fault-tolerant cat state is formed of a single line of qubits exhibiting 1D connectivity. This line of qubits can be carved out of a general graph to connect any pair of qubits. For example, the line of qubits can be any qubit path in a square lattice. Alternatively, the line of qubits can be extended through multiple levels of a tree or through any connectivity line in the entire general graph. The cat state is produced and tested using a post-selection method in conjunction with a small-depth circuit with a maximum depth of 4+2t in one implementation, where t is defined as the fault tolerance of the circuit. In another implementation, referred to herein as a "no-rejection circuit," error correction is performed instead of a technique known as postselection (described with respect to FIG. 3). In this implementation, the cat state can be prepared using a 4+4t maximally deep circuit.

[0007]

[0022] As used herein, "circuit depth" refers to the number of time-separated measurements or rounds of measurements that are performed. For example, a circuit of depth 3 performs three rounds of measurements, each of which may entail multiple simultaneous measurements of individual qubits. In general, the depth of the circuits disclosed herein (e.g., a depth of 4+2t or 4+4t) is significantly less than the typical circuit depth required to implement the existing cat state preparation methodologies described above, which entail measuring random pairs of qubits in a cat state. In these existing cat state preparation methods, the number of measurement rounds (and therefore the computation time) increases with the size of the cat state because each qubit in the cat state moves one at a time from where the cat state is prepared to where the cat state is used to perform data qubit measurements. In the implementations disclosed below with respect to Figures 1-6, the circuit depth of the various exemplary cat state preparation circuits is independent of qubit connectivity, cat state size, and distance between qubits in the set of qubits used to create the cat state.

[0008]

[0023] 1 shows a quantum computing system 100 including a cat state preparation circuit 104 for preparing a cat state used by a measurement circuit 106 to measure two or more groups of qubits in a qubit register 108. According to one implementation, the cat state preparation circuit 104 is a low-depth circuit with a length of 4+2t, where t is the fault tolerance of the measurement circuit 106.

[0009]

[0024] Each cat state prepared by the cat state preparation circuit 104 is “fault-tolerant”, meaning that the prepared cat state contains fewer faults than t, the predefined fault tolerance of the measurement circuit 106. The cat state preparation circuit 104 is designed to generate fault-tolerant cat states by performing a series of operations, described in detail below, to entangle selected qubits in the qubit register 108. The measurement circuit 106 uses the prepared cat states as ancillary qubits to extract measurements of data qubits present in the quantum register 108, meaning that the quantum states of the data qubits are preserved while the ancillary qubits are measured and their quantum states destroyed.

[0010]

[0025] Quantum computing system 100 includes a controller 102 that performs computations by manipulating qubits in a quantum register 108. The qubit values ​​resulting from such manipulations are read by a readout device 114, which includes a cat state preparation circuit 104 and a measurement circuit 106. Each of these circuits may be understood as including hardware, such as one or more quantum and / or classical processors, and / or classically implemented software. The hardware components of cat state preparation circuit 104 and measurement circuit 106 are designed to perform quantum operation sequences on the physical qubits in qubit register 108, and the software components of the two circuits are designed to classically manipulate the measured qubit values.

[0011]

[0026] To measure any combination of two or more data qubits in the qubit register, the controller 102 instructs the cat state preparation circuit 104 to prepare a cat state that entangles together the lines of qubits that provide the physical connectivity between the particular data qubits that are the targets of the measurement. For example, to measure exemplary target qubits 116, 118, 120 (shown in display 124), a cat state can be formed that entangles five connected qubits in adjacent rows 122. These entangled qubits act as auxiliary qubits that are used as tools to extract the quantum state of the target data qubits. To prepare the cat state, the cat state preparation circuit 104 performs a series of operations to entangle each of a pair of neighboring cat state qubits. A detailed example of initial cat state preparation is discussed below with respect to Figures 2A and 2B.

[0012]

[0027] After preparing the initial cat state by entangling the ancillary qubits (e.g., the five qubits in row 122), the cat state preparation circuit 104 performs testing to ensure that the cat state is fault tolerant such that it is within a predefined sufficiency measure. This cat state testing phase entails performing a joint parity measurement sequence on each pair of ancillary qubits that support the cat state. As used herein, the term "joint parity measurement" is used broadly to refer to measurements commonly understood in the industry as native joint parity measurements or joint parity equivalents. For example, a "joint parity equivalent" could be a two-qubit measurement performed by applying a two-qubit gate (such as CNOT or CZ) followed by a single-qubit measurement. Examples of testing techniques are discussed below with respect to FIGS. 3-6. After performing tests to ensure that the prepared cat state contains a number of faults equal to or less than the fault tolerance of the measurement circuit 106, the measurement circuit 106 connects (entangles) the target qubits 116, 118, 120 with the prepared cat state and extracts measurements of the target qubits 116, 118, 120 from the qubits in the prepared cat state.

[0013]

[0028] In the example shown, a measurement result is extracted by performing a single-qubit measurement of each of the five entangled cat state qubits in row 122 (e.g., recovering one bit of information from a different one of the cat state qubits). The extracted measurement represents the parity of the cat state qubits. For example, the extracted measurement may be 00101, where the parity is "1", indicating that the total number of 1s is even.

[0014]

[0029] The measurements extracted by the measurement circuit 106 can be provided to a decoding unit 126 that implements a quantum error correcting code (QECC) to identify the location of faults that have affected the measurement results. In general, a QECC can correct up to a predefined detectable number of faults. This number is generally referred to as the fault tolerance of the QECC. A QECC that is t-fault tolerant can correct up to “t” faults in a syndrome measurement, provided that the syndrome measurement is performed using t-fault tolerant cat states, where t=(d−1) / 2, and d is the minimum distance of the QECC. Using the fault locations identified by the QECC, the parity of the cat state measurements, and a set of known quantum operations performed to perform the measurements and derive the processed measurement results, the controller 102 can determine the quantum states of the individual target qubits 116, 118, 120.

[0015]

[0030] In particular, the reliability of the data qubit measurements described above depends on the quality of the cat state: to ensure reliability, the cat state must, in one implementation, satisfy a sufficiency measure that ensures that the number of faults present in a prepared cat state is less than or equal to a defined fault tolerance level.

[0016]

[0031] When conventional methods are employed, cat state preparation relies on a high degree of connectivity between qubits. For example, a cat state can be tested via a test methodology that requires connectivity between each ancillary qubit used to test the cat state and each of the individual qubits in the cat state. In these designs, cat state preparation and verification is accomplished by a trade-off between qubit connectivity and processing overhead. When connectivity is limited, the processing overhead is much higher. Similarly, processing overhead can be reduced by more complex and expensive architectures that provide higher connectivity between qubits.

[0017]

[0032] In contrast to these conventional approaches, the cat state preparation circuit 104 implements a methodology that significantly reduces the hardware requirements (e.g., both connectivity and processing time) essential to both preparing and verifying a fault-tolerant cat state.

[0018]

[0033] According to one implementation, the cat state preparation circuit 104 is a small-depth circuit with a length of 4+2m or less, where m is the fault-tolerance level of the measurement circuit 106. For example, if the measurement circuit 106 has a fault-tolerance of t=2, this means that the measurement circuit can perform quantum error correction code (QECC) to correct up to two faults present in any given measurement. In an example where the fault-tolerance level is 2, the cat state preparation circuit 104 has a depth of 8 (4+2(2)), which means that a fault-tolerant cat circuit can be implemented with a total of 8 rounds of measurements or less. In contrast, the conventional cat state preparation techniques mentioned above require much higher qubit connectivity and / or much greater depth (e.g., a depth at least proportional to the cat state size) to ensure that the cat state meets the fault-tolerance of the measurement circuit 106.

[0019]

[0034] 2A illustrates a qubit lattice 200 of a quantum computing device. According to one implementation, a data measurement operation of selected qubits ("target data qubits" 204, 206, 208) is performed by preparing a cat state, connecting the prepared cat state to the target data qubits 204, 206, 208, and extracting a measurement. A cat state is an entangled set of k qubits, where each of the target data qubits 204, 206, 208 is physically connected to at least one of the k qubits of the final cat state. For example, qubits 1, 2, 3 (shown in exploded view 210) can be entangled to form a cat state that is used to measure the target data qubits 204, 206, 208.

[0020]

[0035] 2B illustrates an example of a fault-tolerant cat preparation circuit 202 that prepares a cat state that can be used to measure the target data qubits 204, 206, 208 of FIG. 2A. The final cat state used in this target data qubit measurement is referred to herein as the "prepared cat state." In this example, the prepared cat state includes k qubits (e.g., qubits 1, 3, 5) and is formed via a sequence of operations that includes (1) first entangling N qubits, (2) performing tests to verify the reliability of the cat state, and (3) finally discarding some of the N qubits such that the remaining prepared cat state is reduced from N qubits to k qubits.

[0021]

[0036] The N qubits are jointly measured by the cat preparation circuit 202.

number

[0022]

[0037] In contrast, the even-valued qubits (e.g., 2, 4, ...N-1) play a different role and are referred to herein as cat-assisted qubits. The cat-assisted qubits are connected to each other through ancillary qubits, and the function of the cat-assisted qubits is to support measurements that are performed to test the reliability of the prepared cat state.

[0023]

[0038] Based on the above assumptions, the purpose of the cat preparation circuit 202 is to:

number

[0024]

[0039] To generate a prepared cat state, the cat state preparation circuit 202 performs alternating measurements (e.g., M X) and classically implemented update operations (e.g., M Z ) in the horizontal lines. In FIG. 2B, each of the horizontal lines represents a quantum measurement operation sequence (indicated by a shaded box (e.g., M X , M ZZ )) and the classically implemented measurement update operation (denoted by the unshaded boxes (e.g., U Z , U IX ) can be understood as representing the time axis that defines the time period.

[0025]

[0040] The output of the measurement circuit 202 represents, for each individual qubit, a classical quantum state s f qubits, where qubits are equal to the sum of the quantum measurement results(s) observed for that qubit at the circuit output combined with a correction term representing a correction for accumulated errors tracked in conjunction with the individual measurements performed on the qubit within the circuit. Using this principle of classical quantum states, there exists a set of propagation rules that allow (1) to define a measurement update operation (e.g., any Pauli operation of N qubits) associated with each individual measurement of a measurement sequence for the qubit, where each measurement update operation brings the measurement result to a set value (e.g., an error-free trivial result), and (2) to propagate this set of measurement operations to the end of the cat state preparation circuit 202, so that (3) at any given time, the quantum state of each qubit (1-5) can be defined by the measurement value of that qubit that has been classically updated by the set of propagated measurement update operations. The classically updated measurement results are referred to herein as "processed results," which are measurements of classical quantum states.

[0026]

[0041] In more mathematical terms, a Pauli frame for a set of N qubits is a pair of vectors used to keep track of the accumulation of Pauli operations (P) performed on the qubits:

number

number

number

number

number

number

number

number

[0027]

[0042] Using the above principle, the shaded box (e.g., M X , M ZZ Each of the measurement steps in FIG. 2B , denoted by ( ), can be understood as representing a “processed result,” which is the result of a qubit measurement classically updated by taking into account the frame of the qubit, as defined above.

[0028]

[0043] At a first time t1, the cat state preparation circuit 202 calculates an x-reference (M X Following the single qubit x-based measurement, the cat state preparation circuit 202 performs a single qubit measurement in the z-based (U Z ), perform a classically implemented frame update (at time t2). The classically implemented update applies an N-qubit arbitrary Pauli operation in the z-base, which brings the classical quantum state of each of the N qubits to a positive state (|+>). Thus, the measurement M X (performed at t1) and the classically implemented frame update operation U Z The set of and (executed at t2) collectively act to prepare the relevant qubits (1-5) for a positive state (e.g., by defining and classically applying a measurement update that brings the classical quantum state to the |+> state).

[0029]

[0044] At time t3 (after each of qubits 1, 2, 3, 4, and 5 has been prepared to a positive state), joint parity measurements (M ZZ ) is executed. This is followed at t4 by another classically executed set of frame update operations U IX The U IX is an arbitrary two-qubit Pauli update operation, which is an X-based update in which "I" (the identity matrix) is applied to the first qubit to bring the measurement of the second qubit to a known value. This produces a |00>+|11> state across the two qubits.

[0030]

[0045] At time t5, another set of joint parity measurements in the z-base is performed to entangle the pair of qubits 2 and 3, and to entangle the pair of qubits 4 and 5, respectively. This is followed at time t6 by another set of classically performed measurement update operations U X≧3 and U X≧5 Here, U X≧3 is an arbitrary N-qubit Pauli operation operating on qubits 3, 4, and 5 in the X basis to bring the quantum classical state across all five brought-in qubits to a known value (|11111>+|11111>). Similarly, U X≧5 is an arbitrary N-qubit Pauli operation acting on qubit 5 in the x-reference to bring the quantum classical state across all five qubits to a known value, again (|11111>+|11111>).

[0031]

[0046] The sequence of operations described above serves to prepare an initial cat state of N (1 to 5) qubits. Although not shown with respect to FIG. 2B, the cat state circuitry then undergoes a test or postselection stage to ensure that the cat state satisfies a sufficiency measure. Following the test / postselection, the cat ancillary qubits are discarded (e.g., measured independently to disentangle the cat ancillary qubits from the cat state), leaving a prepared cat state containing only qubits 1, 2, and 3. This cat state is then used in the final data qubit measurement.

[0032]

[0047] The testing of the cat state preparation circuit 202 and the derivation of the final cat state are performed, in one implementation, via the same or similar set of operations as detailed with respect to FIG. 3 below.

[0033]

[0048] FIG. 3 shows a fault-tolerant cat state preparation circuit 300 that includes three stages (a), (b), and (c). The first stage (a) of the fault-tolerant cat state preparation circuit 300 is a cat state preparation stage. In this stage, an initial cat state is prepared, such as in a manner consistent with the above description relating to the operation of the cat state preparation circuit 202 of FIG. 2, using the N qubits of the line. The second stage (b) of the fault-tolerant cat state preparation circuit 300 is a postselection stage. In this stage, a joint parity measurement sequence is performed. This sequence includes performing joint parity measurements (M ZZ ), where the first round of measurements involves joint parity measurements of pairs [q1,q2], [q3,q4], [q5,q6], and [q7,q8], and the second subsequent round of measurements involves joint parity measurements of [q2,q3], [q4,q5], [q6,q7], and [q8,q9]. Following each of the two rounds of joint parity measurements, a post-selection (PS) decision step evaluates the measurement result. If the measurement result is non-trivial ("1"), the cat state is rejected and the entire circuit is repeated. Otherwise, the circuit proceeds to the next measurement round.

[0034]

[0049] Once the second round of step (b) is complete (e.g., all eight joint parity measurements shown result in trivial results), the third step (c) of the fault-tolerant cat state preparation circuit 300 measures the even-valued qubits (e.g., the cat auxiliary qubits in FIG. 2B) in the x-reference, thus corrupting the data, thereby disentangling those qubits from the cat state, leaving the final prepared cat state supported by the odd-valued qubits. By tracing every other qubit in block (c), the correlated errors introduced during the last layer of joint measurements are disposed of.

[0035]

[0050] In the example of Figure 3, the fault-tolerant cat preparation circuit 300 (including stages (a), (b), and (c)) has a total depth of 6 (e.g., there are a total of 6 rounds of measurement operations). In stage (a), there are 3 rounds of measurement, in stage (b) there are 2 rounds of measurement, and in stage (c) there is 1 round of measurement.

[0036]

[0051] As discussed via mathematical proofs below, the design of the fault-tolerant cat preparation circuit 300 (specifically, steps (b) and (c)) is sufficient to ensure that the prepared cat state contains only single qubit faults within a predefined accuracy range. For example, if the probability of observing a fault in a measurement is given by p (p=0.001, 0.0001 or smaller event), then the probability of observing two or more qubit faults is p * p (extremely small). Accordingly, a single qubit failure in the fault-tolerant Cat preparation circuit 300 can be corrected by implementing an error-correcting code that is 1-fault tolerant.

[0037]

[0052] Thus, the depth (6) of the fault-tolerant cat-state ready circuit 300 can be expressed as 4+2t, where "t" refers to the maximum fault tolerance of the circuit (e.g., one fault in this example). Circuit families can be derived as generalizations of Figure 3A, as will be evident from the following proposals and proofs.

[0038]

[0053] Proposal 1.4+2δ depth

number

[0039]

[0054] The m-qubit cat state of Proposal 1 is achieved by using a line of qubits with one-dimensional connectivity. In this approach, a cat state can be prepared for any subset of the connected qubits.

[0040]

[0055] Proof of Proposition 1: Since the two Z-errors cancel each other out in the output cat state, it is sufficient to consider the x-error in isolation.

[0041]

[0056] To prove that Fig. 3 is fault-tolerant up to one fault, note that all amplified single faults present at the circuit output are from circuit block (a). These single faults are Pauli errors x i x i+l ...x Nwhere N is the number of qubits in the initial cat state that have a non-trivial X part at the end of stage (a) of the circuit (here the Z part of this error can be neglected). In particular, two x-type errors affecting the measurements return a trivial syndrome. Thus, a single qubit error in the initial cat state can remain undetected in two successive measurements only if at least one of the two measurements contains a fault. This scenario is probabilistically rare. Thus, the second round of measurements in stage (b) effectively serves to catch errors that may be introduced in the first round of measurements in stage (b).

[0042]

[0057] In particular, a single Pauli disturbance of the second measurement of step (b) of the circuit may act on two qubits, but one of these two qubits is disposed of at the end of the circuit, and thus steps (b) and (c) ensure that no new correlation errors are introduced after the preparation of the initial cat state (step (a) of Figure 3).

[0043]

[0058] To reach the circuit output, the Pauli error from block (a) of FIG. 3 must be propagated through the postselection block (b) of FIG. 3 with only trivial measurement results. This requires at least one fault for each pair of consecutive postselection measurements (e.g., the pair formed by measurements 306, 308). Overall, therefore, t additional faults are required to propagate the amplified faults without rejection. This shows that the minimum weight of amplified fault configurations that will not be rejected is at least t+1, proving that the circuit is t-fault tolerant.

[0044]

[0059] Since the maximum weight of a Pauli error affecting the output cat state is [m / 2] up to the stabilizer, we can stop after [m / 2-1] pairs of detection layers if t>[m / 2-1]. At least one failure must occur for the cat state to be rejected. The union bound provides a bound on the rejection probability.

[0045]

[0060] The measurement update of the third level 310 of block (a) involves many qubits. This update can diffuse errors, but only X-type errors are amplified. This is why only Z-type measurements are used in the postselection block.

[0046]

[0061] When the cat state preparation is part of a fault-tolerant scheme based on a stabilizer code with a minimum distance d, a value of t=(d±1) is recommended. Thus, for a bounded cat state preparation, a constant depth t fault-tolerant circuit can be obtained, which is particularly suitable for measuring stabilizer generators of quantum LDPC codes.

[0047]

[0062] The circuits shown in Figures 4A-4C further illustrate a family of fault-tolerant circuits that can be developed from the principles described above.

[0048]

[0063] Figures 4A, 4B, and 4C show a family of cat prepare circuits 400, 402, 404, each of which is t fault tolerant and has a depth of 4+2t. Specifically, the cat prepare circuit 400 of Figure 4A has a fault tolerance of 1 and a depth of 4+2(1)=6. In contrast, the cat prepare circuit 402 of Figure 4B has a fault tolerance of 2 and a depth of 4+2(2)=8. The cat prepare circuit 404 of Figure 4C has a fault tolerance of 3 and a depth of 4+2(3)=10.

[0049]

[0064] Each of the three circuits in Figures 4A, 4B, and 4C shows the three stages discussed above with respect to Figure 3, including (a) initial cat state preparation, (b) postselection, and (c) deentangling the even-valued qubits from the final prepared cat state. Circuits 4A, 4B, and 4C are identical except for the variation in the number of rounds of measurement and postselection in stage (b), which accounts for the resulting variation in the fault tolerance provided by each circuit.

[0050]

[0065] The "post-selection" stage (stage b) improves the quality of the states produced by the circuit. However, if the rejection rate is large (when measurements yield non-trivial results, which causes rejections and restarts the circuit), the initial cat state preparation stage (a) can be performed multiple times before continuing. In this regime, instead of restarting the preparation from scratch, it may be more efficient to try to classically correct the measurement quantum states measured in stage (b) that have non-trivial results. Figure 5 shows a family of circuits that classically correct errors using error correction in the initial cat state, instead of rejecting the state and starting over for each of the non-trivial measurement results of stage (b). This family of circuits can thus be understood as a "no rejection" version of the family introduced in Figures 3 and 4.

[0051]

[0066] 5A-5C show a family of rejection-free fault-tolerant cat state preparation circuits 502, 504, 506, each providing a fault tolerance t and of depth 4+4t. Like the circuits discussed above with respect to FIGS. 3 and 4A-4C, each of the rejection-free fault-tolerant cat state preparation circuits 502, 504, 506 includes three stages (a), (b) and (c). The first stage (a) is an initial cat state preparation stage, which includes the same or similar operations as discussed above with respect to FIGS. 2-4. The second stage (b) is a test and classical correction stage, which includes the same types of measurements (M) as those used in the postselection stage (b) of FIGS. 3 and 4. ZZ ), but the post-selection (PS) rectangle is replaced by a correction rectangle (e.g., correction rectangle 510 in FIG. 5A) at the end of step (b) in FIGS. 5A-5C. Moreover, the measurement rectangle (e.g., rectangle 512) in step (b) includes a joint parity measurement (M ZZ ) and classical corrections (N-qubit Pauli operators) that can be applied to the quantum state of one or more qubits to transform a measurement result from non-trivial to trivial.

[0052]

[0067] Referring to FIG. 5A, the final correction rectangle 510 of step (b) is the sum of all the joint parities (M ZZ ) is an N-qubit Pauli operation that depends on the output of the measurements. Correction rectangle 510 receives as input both the results of the measurements performed in (1) step (b) and the calculated classical corrections (e.g., N-qubit Pauli operations) associated with each measurement result.

[0053]

[0068] Each of the rejection-free fault-tolerant cat-state preparation circuits of 502 in Figure 5A, 504 in Figure 5B, and 506 in Figure 5C is t-fault tolerant and has a depth of 4+4t. This is because the test phase of these circuits (phase (b)) involves twice as many measurements as the postselection phase of the circuits shown and described with respect to Figures 3 and 4. This double measurement in phase (b) allows for an accurate determination of the classical correction to apply at the end of the phase (e.g., correction rectangle 510).

[0054]

[0069] Proposal 2.4+4δ depth

number

[0055]

[0070] Proof of Proposal 2: This proof relies on the same baseline argument as the proof for Proposal 1 above. That is, for an error from stage (a) of the circuit to remain undetected through stage (b), there must be at least one fault per pair of consecutive measurement levels. The goal of the correction rectangle 510 is to identify the residual error induced by any fault configuration with up to t faults. If the correction fails, it means that the residual error E for a fault configuration ω has been confused with the residual error E' for another fault configuration ω'. This combined configuration ω-ω' contains up to ω+ω'<2t+1 faults, leading to a trivial result in stage (b) and a non-trivial residual Pauli error in the output state. This scenario cannot occur when 4t measurement layers are used in stage (b).

[0056]

[0071] 6A-6C show a family of cat state preparation circuits 602, 604, 606, each of which provides fault tolerance t, is 4+2t deep, and each of which has reduced circuit size compared to the similar circuits disclosed with respect to FIGS. 4A-4C. The cat state preparation circuits 602, 604, 606 are obtained by removing unnecessary post-selection measurements from the circuits of FIGS. 4A-4C. The circuit 602 of FIG. 6A is a 1-fault tolerant circuit. The circuit 604 of FIG. 6B is a 2-fault tolerant circuit. The circuit 606 of FIG. 6C is t-fault tolerant for all cases t≧3 in the case of a 9-qubit cat state.

[0057]

[0072] Notably, each measurement of the circuit introduces some additional noise. It is reasonable to expect that the waiting qubits are less erroneous than the measuring qubits. The only difference between the family of circuits of Figs. 6A-6C and the family of circuits of Figs. 4A-4C is that the post-selection measurements involving the first two qubits and the last two qubits of the line are removed. Single faults leading to errors in these qubits are no longer detected. This can be accepted because a single qubit fault in the output state leads to a single qubit Pauli error (depending on the stabilizer), which does not violate the fault tolerance condition. Extending this argument to configurations with two and three faults leads to the circuits of Figs. 4B and 4C.

[0058]

[0073] 7 illustrates example operations 700 for preparing a cat state using a postselection technique in a circuit of fault tolerance t and depth 4+2t. A prepare operation 702 prepares an initial cat state for a line of qubits. An initialize operation initializes an index for a current round of measurement (m=1) that is about to begin. A decide operation 706 starts a measurement round in response to determining that m is still less than or equal to twice the circuit fault tolerance level.

[0059]

[0074] A selection operation 708 selects a pair of neighboring quantum bits from the line, and a measurement operation performs a joint parity measurement in the z-base on the selected pair. A decision operation 712 evaluates the measurement result to determine whether it is non-trivial. If the measurement result is non-trivial ("1"), this indicates that there is a fault in the prepared cat state, and a reject operation 718 rejects the cat state, in which case the preparation operation 702 prepares a new cat state, and operations 704, 706, 708, and 710 are repeated.

[0060]

[0075] Conditional on determining operation 712 determining that the measurement result is trivial ("0"), determining operation 714 determines whether there are any remaining pairs of neighboring qubits for which measurement operation 710 has not yet been performed in the current round of measurement (m=1 for the first round, m=2 for the second round, etc.). If multiple pairs of neighboring qubits remain unmeasured, selection operation 708 selects a new pair of neighboring qubits. Operations 710, 712, 718, etc. are repeated until all neighboring pairs have been measured in the current round "m" and all such measurements have yielded trivial results. Once this condition is met, an incrementation operation increments the measurement round number "m", thereby beginning a new measurement round. The operations described above are repeated until all pairs of bits are measured in a second round (e.g., m=2) and all such measurements again yield trivial results. Notably, the implementation shown in Figures 6A-6C does not perform joint parity measurements on all pairs of neighboring qubits, but only on a subset of those qubits (forming the triangular circuit pattern of Figures 6A-6C). In particular, joint parity measurements can be omitted for several pairs of neighboring qubits because faults that occur at the edge of the circuit cannot propagate to the bulk of the circuit. As a result, these faults are less damaging than faults in the bulk of the circuit, and measurements of these faults are not necessary to detect and correct faults in the bulk of the circuit.

[0061]

[0076] The above process is repeated for "m" rounds of measurements up to twice the fault tolerance level of the circuit. For example, if the circuit has a fault tolerance of 1, then operation 700 performs a total of 2 rounds of joint parity measurements for all pairs of neighboring qubits. Similarly, if the circuit has a fault tolerance of 2, then operation 700 performs a total of 4 rounds of joint parity measurements for all pairs of neighboring qubits.

[0062]

[0077] Once determine operation 706 determines that the number of measurement rounds performed is equal to twice the fault-tolerance level of the circuit, unentangle operation 720 unentangles all even-valued qubits from the initial cat state, effectively reducing the size of the initial cat state to (N+1) / 2, where N is the number of qubits in the initial cat state. Data measurement operation 722 performs data measurements on two or more target qubits by connecting the resulting cat states to the target qubits and extracting measurements.

[0063]

[0078] 8 illustrates example operations 800 for preparing a cat state without rejection (e.g., without using postselection) in a circuit of fault tolerance t and depth 4+2t. A prepare operation 802 prepares an initial cat state for a line of qubits. An initialize operation initializes an index for a current round of measurement (m=1) that is about to begin. A decide operation 806 begins the measurement round in response to determining that m is still less than four times the circuit fault tolerance level.

[0064]

[0079] A selection operation 808 selects a pair of neighboring qubits from the line, and a measurement operation performs a joint parity measurement in the z-base on the selected pair. A decision operation 812 evaluates the measurement result to determine whether it is non-trivial. If the measurement result is non-trivial ("1"), this indicates that there is a fault in the prepared cat state. In this case, an error correction operation 816 computes and stores a classical correction (an N-qubit Pauli operator) that can be used to modify the stored quantum state of one or more qubits to transform the measurement result from non-trivial to trivial.

[0065]

[0080] Following this (or in all cases where the result of measurement operation 812 is trivial), a decision operation 814 determines whether there are any remaining pairs of neighboring qubits for which measurement operation 810 has not yet been performed in the current round of measurement (m=1 for the first round, m=2 for the second round, etc.). If multiple pairs of neighboring qubits remain unmeasured, a selection operation 808 selects a new pair of neighboring qubits. Operations 810, 812, 816, 814 are repeated until all neighboring pairs have been measured in the current round "m" and all such measurements have yielded trivial results. Once this condition is met, an incrementation operation increments the measurement round number "m", thereby initiating a new measurement round. The operations described above are repeated until all pairs of neighboring qubits have been measured in the second round (e.g., m=2) and all such measurements again have yielded trivial results.

[0066]

[0081] The above process is repeated for "m" rounds of measurements up to four times the fault tolerance level of the circuit. For example, if the circuit has a fault tolerance of 1, then operations 800 perform a total of four rounds of joint parity measurements and error correction calculations for all pairs of neighboring qubits. Similarly, if the circuit has a fault tolerance of 2, then operations 800 perform a total of eight rounds of joint parity measurements and error correction calculations for all pairs of neighboring qubits.

[0067]

[0082] Once the determine operation 806 determines that the number of measurement rounds performed is equal to four times the fault tolerance level of the circuit, a classical error correction operation 818 corrects the stored classical quantum state for each qubit based on the error correction operators computed and stored by the error correction calculation 816. An unentangle operation 820 unentangles all even-valued qubits from the initial cat state, effectively reducing the size of the initial cat state to (N+1) / 2, where N is the number of qubits in the initial cat state. A data measurement operation 822 performs data measurements on two or more target qubits by connecting the resulting cat states to the target qubits and extracting measurements.

[0068]

[0083] 9 illustrates an exemplary system for implementing the disclosed technology, including a general purpose computing device in the form of an exemplary conventional PC 900, including one or more processing units 902, a system memory 904, and a system bus 906 that couples various system components including the system memory 904 to the one or more processing units 902. The system bus 906 can be any of several types of bus structures including a memory bus or memory controller, a peripheral bus, and a local bus using any of a variety of bus architectures. The exemplary system memory 904 includes a read only memory (ROM) 908 and a random access memory (RAM) 1210. A basic input / output system (BIOS) 912, containing the basic routines that help transfer information between elements within the PC 900, is stored in the ROM 908.

[0069]

[0084] In an implementation, the system memory 904 stores logical operations for preparing a cat state, such as those described by operation 700 of FIG. 7 and operation 800 of FIG.

[0070]

[0085] The exemplary PC 900 further includes one or more storage devices 930, such as a hard disk drive for reading from and writing to a hard disk, a magnetic disk drive for reading from and writing to a removable magnetic disk, and an optical disk drive (such as a CD-ROM or other optical media) for reading from and writing to a removable optical disk. Such storage devices may be connected to the system bus 1206 by a hard disk drive interface, a magnetic disk drive interface, and an optical drive interface, respectively. The drives and their associated computer-readable media provide non-volatile storage of computer-readable instructions, data structures, program modules, and other data for the PC 1200. The exemplary operating environment may also use other types of computer-readable media capable of storing data accessible by a PC, such as magnetic cassettes, flash memory cards, digital video disks, CDs, DVDs, RAM, ROM, and the like.

[0071]

[0086] A number of program modules, including an operating system, one or more application programs, other program modules, and program data, may be stored in the storage device 930. Decoding logic may be stored in the storage device 930 similar to or in addition to the memory 904. A user may enter commands and information into the PC 900 through one or more input devices 940, such as a keyboard and a pointing device such as a mouse. Other input devices may include a digital camera, a microphone, a joystick, a game pad, a satellite dish, or a scanner, or the like. These and other input devices are often connected to the processing unit(s) 902 through a serial port interface coupled to the system bus 906, but may also be connected by other interfaces, such as a parallel port, a game port, or a universal serial bus (USB). A monitor 946 or other type of display device is also connected to the system bus 906 through an interface, such as a video adapter. Other peripheral output devices 945, such as speakers and a printer (not shown), may also be included.

[0072]

[0087] The PC 900 can operate in a networked environment using logical connections to one or more remote computers, such as the remote computer 960. In some examples, one or more network or communication connections 950 are included. The remote computer 960 can be another PC, a server, a router, a network PC or a peer device or other shared network node, and typically includes many or all of the elements described above with respect to the PC 900, although only a memory storage device 962 is shown in FIG. 12. The personal computer 900 and / or the remote computer 960 can be connected to logical local area networks (LANs) and wide area networks (WANs). Such networked environments are commonplace in offices, enterprise-wide computer networks, intranets and the Internet.

[0073]

[0088] When used in a LAN networking environment, the PC 1200 is connected to the LAN through a network interface. When used in a WAN networking environment, the PC 1200 typically includes a modem or other means for establishing communications over the WAN, such as the Internet. In a networked environment, program modules depicted relative to the personal computer 1200, or portions thereof, may be stored in the remote memory storage device or other locations on the LAN or WAN. The network connections shown are exemplary and other means of establishing a communications link between the computers may be used.

[0074]

[0089] An example method disclosed herein provides for performing a sequence of operations to prepare a cat state in a quantum circuit having a fault tolerance t and a circuit depth of 4+4t or less. The sequence of operations includes at least performing a joint parity measurement sequence on each pair of neighboring qubits of a set of qubits entangled to form an initial cat state, repeating the measurement sequence for at least t rounds, and disentangling a first set of alternating qubits from the initial cat state to form a prepared cat state with a remaining second set of alternating qubits. The second set of alternating qubits is entangled with the first set of alternating qubits along a one-dimensional connectivity line, and the prepared cat state is guaranteed to contain no more than t faults with a predefined probability.

[0075]

[0090] An example method according to any of the preceding methods further involves additional operations for extracting measurements of two or more data qubits using the prepared cat state, the additional operations including entangling the prepared cat state with the two or more data qubits and extracting measurements of the two or more data qubits from the prepared cat state.

[0076]

[0091] In another example method according to any of the preceding methods, the method further includes: (1) in response to determining that any one of the joint parity measurements results in a non-trivial outcome, rejecting the cat state and preparing a new cat state; and (2) in response to determining that none of the joint parity measurements results in a non-trivial outcome, proceeding to a next round of at least t rounds of measurements.

[0077]

[0092] In yet another example method according to any of the preceding methods, repeating the measurement sequence for at least t rounds comprises: * and performing t rounds of the measurement sequence. * t rounds are sufficient to ensure fault-tolerance for up to t failures under any circumstances without rejecting the cat state.

[0078]

[0093] An example of a further alternative method according to any of the preceding methods is * For each of the t rounds of measurements, further computing and storing an N-qubit Pauli operator that can be used to modify the stored quantum states of one or more qubits to transform the measurement result from non-trivial to trivial.

[0079]

[0094] In yet another embodiment of the method according to any of the preceding methods, the method comprises: * Further comprising, following the t rounds, correcting the stored quantum state of each qubit of the set of qubits based on the computed and stored N-qubit Pauli operator.

[0080]

[0095] In yet another method example according to any of the preceding methods, the method further includes preparing a prepared cat state by performing operations to entangle a set of quantum bits, the operations including: (1) performing an x-based measurement of each quantum bit of the set of quantum bits; (2) for each measurement in the x-based having a result that is non-trivial, performing a measurement update operation to invert the result; and (3) performing a joint parity measurement in the z-based for each respective pair of neighboring quantum bits of the set of quantum bits, and for each joint parity measurement in the z-based having a result that is non-trivial, performing a measurement update operation to invert the result.

[0081]

[0096] In yet another example method according to any of the preceding methods, the depth of the quantum circuit is independent of the qubit connectivity, the size of the prepared cat state, and the distance between the qubits of the string of qubits.

[0082]

[0097] An example of a quantum device disclosed herein includes a cat state preparation circuit having a fault tolerance of t and a circuit depth of 4+4t or less. The cat state preparation circuit is configured to prepare a cat state, i.e., to perform a joint parity measurement sequence on each pair of neighboring qubits of a set of qubits entangled to form an initial cat state, to repeat the measurement sequence for at least t rounds, and to disentangle a first set of alternating qubits from the initial cat state to form a prepared cat state that is guaranteed to have t or less faults with a predefined probability. The prepared cat state is formed by the remaining second set of alternating qubits, which are entangled with the first set of alternating qubits along a one-dimensional connectivity line.

[0083]

[0098] In another quantum device according to any preceding quantum device, the cat state preparation circuitry is further configured to perform operations for extracting measurements of two or more data qubits using the prepared cat state, the operations including entangling the prepared cat state with the two or more data qubits and extracting measurements of the two or more data qubits from the prepared cat state.

[0084]

[0099] In another quantum device according to any preceding quantum device, the cat state preparation circuitry is further configured to, in response to determining that any one of the joint parity measurements results in a non-trivial outcome, reject the cat state and prepare a new cat state, and in response to determining that none of the joint parity measurements results in a non-trivial outcome, proceed to a next round of at least t rounds of measurements.

[0085]

[0100] In a further quantum device according to any preceding quantum device, the cat state preparation circuits are collectively sufficient to ensure fault tolerance of up to t faults without rejecting the cat state under any circumstances. * The measurement sequence is repeated for at least t rounds of measurements by performing t rounds of the measurement sequence.

[0086]

[0101] In yet another quantum device according to any of the preceding quantum devices, the cat state preparation circuit includes: * For each of the t rounds of measurements, it is further configured to compute and store an N-qubit Pauli operator, which is an operator that can be used to modify the stored quantum states of one or more qubits to transform the measurement result from non-trivial to trivial.

[0087]

[0102] In another quantum device according to any preceding quantum device, the cat state preparation circuit is further configured to correct the stored quantum state of each qubit of the series of qubits based on the computed and stored N-qubit Pauli operator.

[0088]

[0103] In yet another quantum device according to any preceding quantum device, the cat state preparation circuit is further configured to prepare a prepared cat state by performing operations to entangle a series of qubits, the operations including performing an x-based measurement of each qubit of the series of qubits, performing a measurement update operation to invert each result of the x-based measurement having a non-trivial outcome, performing a joint parity measurement in the z-based for each respective pair of neighboring qubits of the series of qubits, and performing a measurement update operation to invert each result of the joint parity measurement in the z-based having a non-trivial outcome.

[0089]

[0104] In another quantum device according to any of the preceding quantum devices, the depth of the cat state preparation circuitry is independent of the qubit connectivity, the size of the prepared cat state and the distance between the qubits of the string of qubits.

[0090]

[0105] An example of a tangible computer-readable storage medium disclosed herein stores processor-executable instructions for executing a computer process to prepare a cat state in a quantum circuit having a fault tolerance t and a circuit depth of 4+4t or less. The computer process includes performing a joint parity measurement sequence on each pair of neighboring qubits of a set of qubits entangled to form an initial cat state, repeating the measurement sequence for at least t rounds, and disentangling a first set of alternating qubits from the initial cat state to form a prepared cat state with a remaining second set of alternating qubits. The second set of alternating qubits is entangled with the first set of alternating qubits along a one-dimensional connectivity line. The prepared cat state is guaranteed to contain less than t faults with a predefined certainty.

[0091]

[0106] In another computer-readable storage medium example according to any of the preceding storage media, the computer process further includes, in response to determining that any one of the joint parity measurements results in a non-trivial outcome, rejecting the cat state and preparing a new cat state, and in response to determining that none of the joint parity measurements results in a non-trivial outcome, proceeding to a next round of at least t rounds of measurements.

[0092]

[0107] In another example of a computer readable storage medium according to any of the preceding storage media, a computer process includes: * and performing t rounds of the measurement sequence. * t rounds are sufficient to guarantee fault-tolerance for up to t failures with a predefined probability, without rejecting the cat state under any circumstances.

[0093]

[0108] In another example of a computer readable storage medium according to any of the preceding storage media, a computer process includes: *For each of the t rounds of measurements, further computing and storing an N-qubit Pauli operator that can be used to modify the stored quantum states of one or more qubits to transform the measurement result from non-trivial to trivial.

[0094]

[0109] An example of a system d disclosed herein includes a means for performing a sequence of operations to prepare a cat state in a quantum circuit having a fault tolerance t and a circuit depth of 4+4t or less. The system includes at least a means for performing a joint parity measurement sequence on each pair of neighboring qubits of a set of qubits entangled to form an initial cat state, a means for repeating the measurement sequence for at least t rounds, and a means for disentangling a first set of alternating qubits from the initial cat state to form a prepared cat state with a remaining second set of alternating qubits. The second set of alternating qubits is entangled with the first set of alternating qubits along a one-dimensional connectivity line, and the prepared cat state is guaranteed to contain no more than t faults with a predefined probability.

[0095]

[0110] The above specification, examples and data provide a complete description of the structure and use of the exemplary implementations. Many implementations can be made without departing from the spirit and scope of the invention as claimed, and the following appended claims define the invention. Moreover, the structural features of different examples can be combined in further alternative implementations without departing from the scope of the claims as described. The above specification, examples and data provide a complete description of the structure and use of the exemplary implementations. Many implementations can be made without departing from the spirit and scope of the invention as claimed, and the following appended claims define the invention. Moreover, the structural features of different examples can be combined in further alternative implementations without departing from the scope of the claims as described.

Claims

1. 1. A method comprising: performing a sequence of operations to prepare a cat state in a quantum circuit having a fault tolerance t and a circuit depth less than or equal to 4+4t, the sequence of operations comprising: performing a joint parity measurement sequence on each pair of neighboring qubits of the set of qubits that were entangled to form the initial cat state; repeating said measurement sequence for at least t rounds; disentangling a first set of alternating qubits from the initial cat state to form a prepared cat state guaranteed to contain no more than t faults with a predefined degree of certainty, the prepared cat state being formed by a second set of alternating qubits that are the remainder of the string of qubits, the second set of alternating qubits being entangled with the first set of alternating qubits along one-dimensional connectivity lines; The method includes at least the following steps.

2. performing an additional operation to extract measurements of two or more data qubits using the prepared cat state.

2. The method of claim 1, further comprising the additional action of: entangling the prepared cat state with the two or more data qubits; extracting measurements of the two or more data qubits from the prepared cat state; and A method comprising:

3. in response to determining that any one of the joint parity measurements yields a non-trivial result, rejecting the cat state and preparing a new cat state; in response to determining that none of the joint parity measurements yields a non-trivial result, proceeding to a next round of the at least t rounds of measurements; The method of claim 1 further comprising:

4. Repeating the measurement sequence for at least t rounds; Under any circumstances, the predefined probability is sufficient to guarantee fault tolerance for up to t faults without rejecting the cat state. ＊ The method of claim 1 , further comprising performing t rounds of the measurement sequence.

5. 2 ＊ for each of t rounds of measurements, computing and storing an N-qubit Pauli operator that can be used to modify the stored quantum states of one or more of the qubits to transform the measurement results from non-trivial to trivial. The method of claim 4 further comprising:

6. 2 ＊ correcting the stored quantum state of each qubit of the series of qubits based on the computed and stored N-qubit Pauli operator following t rounds. The method of claim 5 further comprising:

7. Preparing the prepared cat state by performing a sequence of operations to entangle the series of qubits.

2. The method of claim 1, wherein the sequence of operations further comprises: performing an x-based measurement of each said qubit of said series of qubits; for each measurement on the x basis having a non-trivial result, performing a measurement update operation to invert the result; performing a joint parity measurement in the z-base for each respective pair of neighboring qubits of the set of qubits; for each joint parity measurement in the z-base having a non-trivial result, performing a measurement update operation to invert the result; A method comprising:

8. 2. The method of claim 1, wherein the depth of the quantum circuit is independent of qubit connectivity, the size of the prepared cat state, and the distance between the qubits of the series of qubits.

9. Cat state preparation circuit with fault tolerance t and circuit depth less than or equal to 4+4t A quantum device comprising: performing a joint parity measurement sequence on each pair of neighboring qubits of the set of qubits that were entangled to form the initial cat state; repeating said measurement sequence for at least t rounds; disentangling a first set of alternating qubits from the initial cat state to form a prepared cat state guaranteed to contain no more than t faults with a predefined degree of certainty, the prepared cat state being formed by a second set of alternating qubits that are the remainder of the string of qubits, the second set of alternating qubits being entangled with the first set of alternating qubits along one-dimensional connectivity lines; A quantum device configured to:

10. The cat state preparation circuit includes: and performing operations to extract measurements of two or more data qubits using the prepared cat state, the operations comprising: entangling the prepared cat state with the two or more data qubits; extracting measurements of the two or more data qubits from the prepared cat state; and The quantum device of claim 9 .

11. The cat state preparation circuit includes: in response to determining that any one of the joint parity measurements yields a non-trivial result, rejecting the cat state and preparing a new cat state; in response to determining that none of the joint parity measurements yields a non-trivial result, proceeding to a next round of the at least t rounds of measurements; The quantum device of claim 9 , further configured to:

12. The cat state preparation circuit includes: Under any circumstances, the predefined probability is sufficient to guarantee fault tolerance for up to t faults without rejecting the cat state. ＊ 10. The quantum device of claim 9, wherein the measurement sequence is repeated over the at least t rounds of measurements by performing t rounds of the measurement sequence.

13. The cat state preparation circuit includes: 2 ＊ 13. The quantum device of claim 12, further configured to compute and store, for each of t rounds of measurements, an N-qubit Pauli operator, the N-qubit Pauli operator usable to modify the stored quantum states of one or more of the qubits to transform the measurement results from non-trivial to trivial.

14. The cat state preparation circuit includes: correcting the stored quantum state of each qubit of the series of qubits based on the computed and stored N-qubit Pauli operator. The quantum device of claim 13 further configured as follows:

15. The cat state preparation circuit includes: and preparing the prepared cat state by performing a sequence of operations for entangling the set of qubits, the sequence of operations comprising: performing an x-based measurement of each said qubit of said series of qubits; for each measurement on the x basis having a non-trivial result, performing a measurement update operation to invert the result; performing a joint parity measurement in the z-base for each respective pair of neighboring qubits of the set of qubits; for each joint parity measurement in the z-base having a non-trivial result, performing a measurement update operation to invert the result; The quantum device of claim 9 .

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