Fault-tolerant quantum cat state preparation method, device, apparatus and storage medium

Abstract

A quantum computing system adapted 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 by performing a series of operations, the operations comprising: performing a sequence of joint parity measurements on individual adjacent pairs of qubits in an entangled series of qubits to form an initial cat state; repeating the sequence of measurements in at least t rounds; and separating a first set of alternate qubits from the initial cat state, the prepared cat state formed from a second set of alternate qubits remaining, the second set of alternate qubits interleaved with the first set of alternate qubits along a one-dimensional connected line, the series of operations sufficient to guarantee that the prepared cat state has less than or equal to t errors.

Description

Technical Field

[0001] This application relates to the field of quantum computing, and specifically to methods, devices, apparatuses, and storage media for preparing fault-tolerant quantum cat states. Background Technology

[0002] A Shor cat state, or simply "cat state," is a quantum state formed by highly entangled multiple qubits. Cat states are used in a variety of types of quantum operations to provide fault-tolerant computation, including the implementation of logic operations (e.g., operations within and across different logic blocks), quantum error correction, and teleportation quantum states from one bit to another. For example, a cat state can be used to perform Pauli operator measurements on two or more data qubits. Such a measurement requires (1) entanglement of a set of auxiliary qubits through a series of operations to create cat states for these auxiliary qubits; (2) entanglement of the data qubit to be measured with the prepared cat state qubits; and (3) performing a single-qubit measurement on each auxiliary qubit to extract the data qubit measurement, which is given by the collective parity of the bit being measured. Summary of the Invention

[0003] According to one implementation, it has fault tolerance. t The sum is less than or equal to 4+4 t A quantum circuit of circuit depth executes a cat state by performing a series of operations, including: performing a joint parity measurement sequence on individual adjacent qubit pairs in an entangled set of qubits to form an initial cat state; repeating the measurement sequence for at least t rounds; and untangling 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 interleaved with the first set of alternating qubits along a line of one-dimensional connectivity. The prepared cat state is guaranteed to have a predefined degree of determinism, including less than or equal to t The number of errors is used to de-entangle the system. This article also describes and lists other implementations. Attached Figure Description

[0004] Figure 1 The illustration shows a quantum computing system including a short-depth fault-tolerant cat-state preparation circuit.

[0005] Figure 2A The illustration shows a qubit grid in a quantum computing device.

[0006] Figure 2B The diagram illustrates the preparation that can be used for measurement. Figure 2A Example fault-tolerant cat preparation circuit for the target data qubit cat state.

[0007] Figure 3 Another example of a short-depth fault-tolerant cat-state preparation circuit is illustrated.

[0008] Figure 4A The diagram illustrates a depth of 4+2 t Fault tolerance t An exemplary cat-state preparation circuit.

[0009] Figure 4B The diagram illustrates a depth of 4+2 t Fault tolerance t Another exemplary cat-state preparation circuit.

[0010] Figure 4C The diagram illustrates a depth of 4+2 t Fault tolerance t Another exemplary cat-state preparation circuit.

[0011] Figure 5A The diagram illustrates a depth of 4+4 t Fault tolerance t An exemplary no-drop fault-tolerant cat-state preparation circuit.

[0012] Figure 5B The diagram illustrates a depth of 4+4 t Fault tolerance t Another exemplary no-drop fault-tolerant cat-state preparation circuit.

[0013] Figure 5C The diagram illustrates a depth of 4+4 t Fault tolerance t Another example of a no-drop fault-tolerant cat-state preparation circuit.

[0014] Figure 6A The diagram illustrates fault tolerance. t And the depth is 4+2 t of Figures 4A-4C Compared to the smaller size example cat-state preparation circuit.

[0015] Figure 6B The diagram illustrates fault tolerance. t And the depth is 4+2 t of Figures 4A-4C Another example of a cat-state preparation circuit with a smaller size.

[0016] Figure 6C The diagram illustrates fault tolerance. t And the depth is 4+2 t of Figures 4A-4C Another example of a cat-state preparation circuit with a smaller size.

[0017] Figure 7 The diagram illustrates the use of fault tolerance. t and depth 4+2 t The circuit uses a post-selection technique to prepare example operations for the cat state.

[0018] Figure 8 The diagram illustrates the use of fault tolerance. t and depth 4+2 t Prepare example operation 800 in the circuit without discarding the cat state.

[0019] Figure 9 An exemplary computing environment suitable for implementing aspects of the disclosed technology is illustrated. Detailed Implementation

[0020] Preparing a Shor's cat state requires performing a series of measurements to entangle a selected set of qubits. This process can be followed by a post-selection phase, in which a series of measurements are performed to verify that the cat state is ready with a certain degree of sufficiency (e.g., the number of errors contained in the cat state is equal to or less than the predefined fault tolerance of the corresponding measurement circuit).

[0021] Existing post-selection (testing) methods for Shor's cat states measure the parity of random qubit pairs within the cat state. Ensuring sufficiently accurate preparation of the cat state requires a large number of measurements and high qubit connectivity (e.g., connectivity between each auxiliary element used in the post-selection test and all qubits in the cat state). Such high connectivity is difficult to achieve in currently proposed quantum platforms, including, for example, within a qubit grid, using superconducting qubits, Majorana qubits, or logical qubits encoded with surface codes, color codes, or small code blocks. Furthermore, this method for testing random qubit pairs in the cat state requires a large number of measurement rounds, each consuming significant processing time. For example, each measurement round might require approximately 100 ns for superconducting qubits and 100 microseconds for ion traps.

[0022] The technology disclosed herein provides circuitry for cat-state preparation and testing methods for cat-state preparation, which, compared to existing testing methods described above, can be implemented with lower processing time overhead (e.g., fewer measurement cycles) and reduced hardware requirements (e.g., less connectivity between qubits). According to one implementation, the fault-tolerant cat-state is formed by a single row of qubits exhibiting one-dimensional connectivity. This qubit line can be sculpted from a general graph to connect any pair of qubits. For example, the qubit line can be any qubit path in a square grid. Alternatively, the qubit line can extend through multiple levels of a tree or through any connection line in the general graph. The cat-state is generated and tested using a post-selection method combined with short-depth circuitry; in one implementation, the maximum depth is 4+2. t ,in t This is defined as the fault tolerance of a circuit. In another implementation, referred to in this paper as a "drop-out-free circuit," error correction is performed instead of a technique called post-selection (see [reference]). Figure 3 (Description). In this implementation, the cat state can use a maximum depth of 4+4. t The circuit was prepared.

[0023] As used herein, “circuit depth” refers to the number of measurement rounds performed or time-interval measurements. For example, a depth 3 circuit performs three rounds of measurements, each round potentially requiring multiple individual qubit measurements to be performed simultaneously. Typically, the circuits disclosed herein (e.g., depth 4+2) t Or 4+4 t The depth of this method is significantly shorter than the typical circuit depth required to implement the existing cat-state preparation methods described above, which require measuring random qubit pairs in the cat state. In these existing cat-state preparation methods, the number of measurement wheels (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 the position where the cat state is prepared to the position where the data qubit measurement is performed using the cat state. (Further details are provided below.) Figure 1 In the implementations disclosed in -6, the circuit depth of the various exemplary cat-state preparation circuits is independent of the qubit connectivity, the cat-state size, and the distance between the qubits in the series used to create the cat-state.

[0024] Figure 1 The illustration shows a quantum computing system 100 including a cat-state preparation circuit 104 for preparing a cat state, which a measurement circuit 106 uses to measure groups of two or more qubits in a qubit register 108. According to one implementation, the cat-state preparation circuit 104 is of length 4+2. t Short-depth circuitry. Among them... t It is the fault tolerance of the measurement circuit 106.

[0025] Each cat state prepared by the cat state preparation circuit 104 is "fault-tolerant", meaning that the prepared cat state contains fewer errors than the predefined fault tolerance of the measurement circuit 106. t As will be described in detail below, the cat-state preparation circuit 104 is designed to generate a fault-tolerant cat state by performing a series of operations to entangle the group of qubits selected in the qubit register 108. The measurement circuit 106 uses the cat state as an auxiliary qubit preparation, which means measuring the auxiliary qubits and destroying their quantum states while preserving the quantum states of the data qubits.

[0026] The quantum computing system 100 includes a controller 102 that performs computations by manipulating qubits within a quantum register 108. The values ​​of the qubits resulting from this manipulation are read by a readout device 114, which includes a cat-state preparation circuit 104 and a measurement circuit 106. Each of these circuits can be understood to include hardware, such as one or more quantum and / or classical processors, and / or classically implemented software. The hardware components of the cat-state preparation circuit 104 and the measurement circuit 106 are designed to perform a series of quantum operations on the physical qubits in the qubit register 108, while the software components of both circuits are designed to classically manipulate the measured qubit values.

[0027] To measure any combination of two or more data qubits in a qubit register, controller 102 instructs cat-state preparation circuit 104 to prepare a cat state that entangles a line of qubits together, providing a measurement of the physical connection between the target specific data qubits. For example, to measure exemplary target qubits 116, 118, and 120 (shown in view 124), a cat state can be formed to entangle five connected qubits in adjacent rows 122. These entangled qubits act as an aid, serving as a tool for extracting the quantum state of the target data qubits. To prepare the cat state, cat-state preparation circuit 104 performs a series of operations to entangle each adjacent pair of cat-state qubits. (Refer to below) Figure 2A-2B A detailed example of preparing the initial cat state is discussed.

[0028] After preparing the initial cat state by entangled auxiliary qubits (e.g., the five qubits in row 122), the cat state preparation circuit 104 performs tests to ensure that the cat state is fault-tolerant within a predefined sufficiency metric. This cat state testing phase requires performing a sequence of joint parity measurements on the single auxiliary qubit pair supporting the cat state. As used herein, the term "joint parity measurement" is broadly used to refer to a naturally occurring joint parity measurement or a measurement generally understood in the industry as joint parity equivalence. For example, "joint parity equivalence" could be a two-qubit measurement performed by applying two qubit gates (e.g., CNOT or CZ), followed by a single-qubit measurement. Example testing techniques will be discussed below regarding... Figure 3 - Figure 6 discusses this. After testing to ensure that the prepared cat state includes an error number less than or equal to the fault tolerance of the measurement circuit 106, the measurement circuit 106 connects (entangles) the target qubits 116, 118, and 120 with the prepared cat state and extracts the measurements of the target qubits 116, 118, and 120 from the qubits of the prepared cat state.

[0029] In the example shown, the measurement result is extracted by performing a single qubit measurement on each of the five entangled cat-state qubits in row 122 (e.g., retrieving one bit of information from each distinct cat-state qubit). The extracted measurement represents the parity of the cat-state qubit. For example, the extracted measurement could be 0 01 0 1, with parity "1", where "1" indicates that the total number is even.

[0030] The measurement results extracted by measurement circuit 106 can be provided to decoding unit 126, which uses solid-line quantum error-correcting codes (QECC) to identify the error locations affecting the measurement results. Generally, QECC can correct up to a predefined number of detectable errors. This number is often referred to as the fault tolerance of QECC. A fault-tolerant QECC of t can correct up to t errors in a sub-measurement, provided that the sub-measurement is performed using a fault-tolerant cat state of t, where t = (dl) / 2 and d is the minimum distance of the QECC. Using the error locations identified by the QECC, the parity of the cat-state measurement, and a known set of quantum operations performed to execute the measurement and obtain the processed measurement results, controller 102 is able to determine the quantum states of individual target qubits 116, 118, and 120.

[0031] It is worth noting that the reliability of the aforementioned data qubit measurements depends on the quality of the cat state. To ensure reliability, the cat state must satisfy a sufficiency metric, which in one implementation ensures that the number of errors present in the prepared cat state is less than or equal to the defined fault tolerance level.

[0032] When using traditional methods, cat state preparation relies on high connectivity between qubits. For example, cat states can be tested using a test method that requires connectivity between each auxiliary element used to test the cat state and each individual qubit within the cat state. In these designs, cat state preparation and verification are achieved through a trade-off between qubit connectivity and processing overhead. When connectivity is limited, the processing overhead is much higher. Similarly, the processing overhead can be reduced through more complex and expensive architectures that provide greater connectivity between qubits.

[0033] Compared to these traditional methods, the cat-state preparation circuit 104 implements a method that significantly reduces the hardware requirements (e.g., connectivity and processing time) necessary for preparing and verifying fault-tolerant cat states.

[0034] According to one implementation, the cat-state preparation circuit 104 has a length less than or equal to 4 + 2. m Short-depth circuits, in which mThis refers to the fault tolerance level of measurement circuit 106. For example, if measurement circuit 106 has a fault tolerance of t=2, this means that the measurement circuit is capable of executing quantum error-correcting codes (QECC) to correct two errors present in any given measurement. In the example of fault tolerance level 2, the depth of cat state preparation circuit 104 is 8 (4+2(2)), which means that measurements of the fault-tolerant cat circuit can be performed in no more than 8 full rounds. In contrast, the conventional cat state preparation techniques described above require greater qubit connectivity and / or greater depth (e.g., at least a depth proportional to the cat state size) to guarantee that the cat state satisfies measurement circuit 106.

[0035] Figure 2A The illustration shows a qubit grid 200 in a quantum computing device. According to one implementation, a data measurement operation of selected qubits (“target data qubits” 204, 206, and 208) is performed by preparing a cat state, connecting the prepared cat state to the target data qubits 204, 206, and 208, and extracting the measurement. The cat state entangles a series of k qubits, with each target data qubit 204, 206, 208 physically connected to at least one of the k qubits in the final cat state. For example, qubits 1, 2, and 3 (shown in exploded diagram 210) can be entangled to form a cat state for measuring the target data qubits 204, 206, and 208.

[0036] Figure 2B The illustration shows an example fault-tolerant cat preparation circuit 202, whose preparation can be used for measurement. Figure 2A The target data qubits are cat states of 204, 206, and 208. The final cat states used in this target data qubit measurement are referred to herein as “prepared cat states”. In this example, the prepared cat states consist of k qubits (e.g., qubits 1, 3, and 5) and are formed via a series of operations including: (1) initially entangled N qubits; (2) performing tests to verify the reliability of the cat states; and (3) finally discarding some of the N qubits, reducing the remaining prepared cat states from N to k qubits.

[0037] N qubits are physically arranged into lines with 1D connectivity, such that the cat preparation circuit 202 is equipped to perform joint measurements for i = 1, 2, ..., N-1. In the illustrated example (in) Figure 2A(As shown in the decomposed diagram 210), the N-qubit sequence comprises a series of qubits labeled 1, 2, 3, 4, 5, arranged along a curve, intersecting the qubits in numerical order. For example, but not limited to, the total number of qubits in sequence N is assumed to be odd, such that N equals 2n+1, where n is the odd number of qubits in sequence N. The n odd number qubits 1, 3, ... N are considered as auxiliary qubits to form the prepared cat states that can be used to measure the target data qubits 204, 206, 208.

[0038] In contrast, even-numbered qubits (e.g., 2, 4, ..., N-1) serve a different purpose and are referred to in this paper as cat ancilla qubits. Cat ancilla qubits are interconnected by auxiliary qubits, and their function is to support the execution of measurements to test the reliability of prepared cat states.

[0039] Given the aforementioned assumptions, the ultimate goal of the cat preparation circuit 202 is to prepare a k-qubit cat state of the following form:

[0040] (1)

[0041] exist , The cat state is supported on an arbitrary subset of the k data qubits. In the example of Figure 2, qubits 2, 4, 6, 8, and 10 are cat-auxiliary qubits that do not support the final cat state. Instead, the cat state is supported by the set of qubits. support.

[0042] To generate the prepared cat state, the cat state preparation circuit 202 performs a series of operations, including alternating measurements (e.g., M). x ) and the update operation of the classic implementation (e.g., M) z The combination of ) in. Figure 2B In the diagram, each horizontal line can be interpreted as representing a series of quantum measurement operations (defined by the shaded box (e.g., M)). x M zz (represented by) and the classic implementation of the measurement update operation (by an unshaded box (e.g., M) zz () indicates the timeline.

[0043] For each individual qubit, the output of the measurement circuit 202 can be represented as a classical quantum state. This is equal to the sum of the quantum measurement result observed for that qubit at the circuit output and the correction term, which represents the correction for the accumulated error tracked along 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) defining measurement update operations (e.g., arbitrary Pauli operations for N-qubits) associated with each individual measurement in the sequence of measurements on the qubit, where each measurement update operation forces the measurement result to a set value (e.g., a trivial result without error); (2) propagating this set of measurement operations to the end of the cat-state preparation circuit 202 such that (3) the quantum state of each qubit (1-5) can be classically updated at any given time by the measurement of that qubit by the propagation set of measurement update operations. The classical update result of the measurement is referred to herein as the “processed result,” which is the measurement of the classical quantum state.

[0044] In more mathematical terms, a Pauli frame of an N-qubit set can be defined as a pair of vectors. Used to track the accumulation of Pauli operations (P) performed on qubits. The state of a quantum device can be defined as a classical quantum state at any given time. ),in f represents the state of the N-qubit system, and f is the current Pauli frame. Manipulating classical data. This allows for the simple implementation of Pauli gates. The measurement of the Pauli observable Q projects the quantum state onto an image in a projector. (Where I is the identity matrix) and return the processed result:

[0045] (2)

[0046] Where s is the actual measurement result. It corresponds to the Pauli frame, and It is the symplectic representation of Q, with square brackets [.,.] indicating the symplectic inner product. Therefore, post-measurement processing only requires local frames of the measured qubits. The data. Two classical quantum states. This results in the same outcome distribution for any measurement sequence (M1, M2). In other words, any Pauli operation P can be implemented as a frame update:

[0047] (3)

[0048] This provides a classical implementation for any one- or two-dimensional Pauli measurement. The frame is initialized with a zero vector and updated during measurement updates according to equation (3). In summary, the classical implementation of the frame update operation (e.g., U...) z This is achieved using a classic control device, which is capable of:

[0049] (1) Store 2N bits of the Pauli frame f;

[0050] (2) Frame update Calculate bitwise XOR; and

[0051] (3) Processing of results Calculate the inner product of two variables

[0052] Using the aforementioned principle, the shadow box (e.g., M) x M zz ) represents Figure 2B Each of the measurement steps can be understood as representing the "result of the processing," as defined above, which is the result of a classically updated qubit measurement by taking into account the frame of the qubit.

[0053] At the first time point t1, the cat-state preparation circuit 202 performs x-basis (M) on each individual qubit in sequence N(1-5). x Single-qubit measurements in ). After single-qubit x-basis measurements, cat-state preparation circuit 202 (at time t2) performs measurements on each z-basis (U) in the N qubits. z The classical implementation of frame updates is performed within the `<z>` matrix. The classical implementation applies an arbitrary Pauli operation to the `<n>` qubits, forcing the classical quantum state of each `<n>` qubit to be normal. Therefore, measuring M x The set (executed in t1) and the classic implementation of the frame update operation U z (Executed at t2) Co-operation to prepare the relevant qubits of the normal state (1-5) (e.g., by defining and classically applying to force the classical quantum state to be State measurement and update.

[0054] At time t3 (after each qubit 1, 2, 3, 4, 5 is ready to be in the normal state), a z-basis joint parity measurement (M) is performed. zz ) with adjacent qubits 1, 2 and 3, 4 respectively entangled. Then at t4, there is another set of classically implemented frame update operations U. IX U IX It is an arbitrary two-qubit Pauli update operation that applies "I" (identity matrix) to the first qubit and the X basis update to force the measurement of the second qubit to a known value. This will produce on both qubits The state.

[0055] At time t5, another set of z-based joint parity measurements is performed to entangle qubit pairs 2 and 3, and 4 and 5, respectively. At time t6, another set of measurement update operations implemented classically follows. and .here, It is an arbitrary N-qubit Pauli operation that operates on qubits 3, 4, and 5 in the X basis to force the quantum classical states on all five qubits to known values ​​(|11111>+111111>). Similarly, It is an arbitrary N-qubit Pauli operation that operates on qubit 5 in the x basis to force the quantum classical state on all five qubits to be equal to the known value again (|11111>+|11111>).

[0056] The above sequence of operations is used to prepare the initial cat state for the N(1-5) qubits. Although there is no information regarding... Figure 2B As shown, the cat-state circuit then undergoes a test or post-selection phase to ensure that the cat states satisfy a sufficiency metric. After the test / post-selection, the cat-assistant qubits are discarded (e.g., independently measured to de-entangle them with the cat states), leaving the prepared cat states, which consist only of qubits 1, 2, and 3. This cat state is then used for the final data qubit measurement.

[0057] In one implementation, the testing of the cat-state preparation circuit 202 and the derivation of the final cat state are performed by referring to... Figure 3 A set of operations that are identical or similar to the detailed operations are performed.

[0058] Figure 3 The diagram illustrates a fault-tolerant cat-state preparation circuit 300, comprising three stages (a), (b), and (c). The first stage (a) of the fault-tolerant cat-state preparation circuit 300 is the cat-state preparation stage. In this stage, an initial cat state is prepared using N qubits in a row, for example, in a manner consistent with the description above regarding the operation of the cat-state preparation circuit 202 in Figure 2. The second stage (b) of the fault-tolerant cat-state preparation circuit 300 is the post-selection stage. In this stage, a sequence of joint parity measurements is performed. This sequence requires performing a z-basis joint parity measurement (M) on each pair of adjacent qubits in the sequence. zz (The first round of measurements includes joint parity measurements of [q1, q2], [q3, q4], [q5, q6], and [q7, q8], while the second round includes joint parity measurements of [q2, q3], [q4, q5], [q6, q7], and [q8, q9].) After each of the two rounds of joint parity measurements, a post-selection (PS) decision step evaluates the measurement result. If the measurement result is nontrivial ("1"), the cat state is discarded and the entire circuit is repeated. Otherwise, the circuit continues to the next round of measurements.

[0059] Upon completion of the second phase (b) (e.g., when the joint parity measurement of all 8 diagrams has resulted in a trivial result), the third phase (c) of the fault-tolerant cat-state preparation circuit 300 measures even-numbered quantum bits in the x-basis (e.g., Figure 2BThe cat-assisted state (i.e., the data is corrupted, thus unentangled these qubits with the cat state, leaving the final, ready cat state supported by an odd number of qubits) is obtained. By tracking all other qubits in block (c), the relevant errors introduced during the final layer of joint measurements can be discarded.

[0060] exist Figure 3 In the example, the fault-tolerant cat preparation circuit 300 (including stages (a), (b), and (c)) has a total depth of 6 (e.g., a total of six rounds of measurement operations). In stage (a), there are three rounds of measurement; in stage (b), there are two rounds of measurement; and in stage (c), there is one round of measurement.

[0061] As will be discussed below with mathematical proof, the fault-tolerant cat preparation circuit 300 (specifically, stages (b) and (c)) is designed sufficiently to guarantee that the prepared cat state includes only a single qubit error within a predefined range of determinism. For example, if the probability of observing an error in a measurement is given by p, where p = 0.001, 0.0001, or less, the probability of observing two or more qubit errors is approximately p*p (extremely small). A single qubit error in the fault-tolerant cat preparation circuit 300 can be corrected by implementing a 1-fault-tolerant error-correcting code.

[0062] The depth (6) of the fault-tolerant cat-state preparation circuit 300 can therefore be expressed as 4+2t, where “t” refers to the maximum fault tolerance of the circuit (e.g., 1 fault in this example). A series of circuits can be derived as Figure 3 The generalization of A is proved by the following propositions and proofs:

[0063] Proposition 1. It is possible to use m-1 auxiliary qubits and depths of... (in )of A fault-tolerant circuit t prepares an m-qubit cat state on a row of qubits. The discard probability of the circuit is proportional to Ntp, where p is the probability of dropping the state. The maximum error probability above the error location.

[0064] The m-qubit cat state of Proposition 1 is realized by using a row of qubits with one-dimensional connectivity. In this method, the cat state is prepared on an arbitrary subset of the connected qubits.

[0065] Proof of Proposition 1: Since the two Z errors on the output cat state cancel each other out, it is sufficient to consider the x error alone.

[0066] To prove Figure 3 The circuit is fault-tolerant to a maximum of one error. Note that all amplified individual errors present in the circuit output originate from circuit block (a). These individual errors are referred to as Pauli errors. The propagation is performed, where N is the number of qubits in the initial cat state, with a nontrivial X component at the end of stage (a) of the circuit. (The Z component of this error can be ignored here). It is worth noting that two x-type errors affecting the measurement return a trivial syndrome. Therefore, a single qubit error in the initial cat state is likely to go undetected in two consecutive measurements only if at least one of the two measurements contains an error. This situation is probabilistically rare. Therefore, the second round of measurements in stage (b) is effectively used to capture the error that may have been introduced in the first round of measurements in stage (b).

[0067] It is worth noting that a single Pauli error in the measurement of the second stage (b) of the circuit can act on two qubits, but one of these two qubits is discarded at the end of the circuit; therefore, stages (b) and (c) ensure that no new relevant errors are introduced after the preparation of the initial cat state. Figure 3 Phase (a) in the middle.

[0068] To achieve the circuit output, from Figure 3 The Pauli error of block (a) must be obtained through only trivial measurement results. Figure 3 The propagation of the post-selection block (b) is as follows. For each pair of consecutive post-selection measurements (e.g., a pair formed by measurements 306, 308), this requires at least one error. Overall, therefore, t additional errors are needed to propagate the amplified error without it being discarded. This shows that the minimum weight of the non-discarding amplified error configuration is at least t+1, proving that the circuit is t It is forgiving.

[0069] If t > [m / 2 - 1], the detection layer can be stopped at [m / 2 - 1] pairs until the stabilizer, because the maximum weight of the Pauli error affecting the output cat state is [m / 2]. For a cat state to be discarded, at least one error must occur. The joint boundary provides the boundary for the discard probability.

[0070] The measurement update at level 310 of block (a) involves a large number of qubits. Although this update may propagate errors, it will only amplify X-type errors. This is why only Z-type measurements are used in the later-selected block.

[0071] When cat-state preparation is part of a fault-tolerant scheme based on a stabilizer code with a minimum distance d, the value t = (d ± 1) is recommended. Therefore, a constant depth t fault-tolerant circuit can be obtained for bounded cat-state preparation. This is particularly relevant to stabilizer generators for measuring quantum LDPC codes.

[0072] Figures 4A-4C The circuit shown further illustrates the family of fault-tolerant circuits that can be developed from the aforementioned principles.

[0073] Figure 4A , Figure 4B and Figure 4C The diagram illustrates a series of cat-preparation circuits 400, 402, and 404, each with t-tolerance and a depth of 4+2t. Specifically, Figure 4A The cat preparation circuit 400 has a fault tolerance of 1 and a depth of 4 + 2(1) = 6. In contrast, Figure 4B The cat preparation circuit 402 has a fault tolerance of 2 and a depth of 4+2(2)=8. Figure 4C The cat preparation circuit 404 has a fault tolerance of 3 and a depth of 4+2(3)=10.

[0074] Three Figure 4A , Figure 4B and Figure 4C Each of the symbols in the circuit represents the above about Figure 3 The three phases discussed include (a) initial cat-state preparation; (b) post-selection; and (c) de-entanglement of an even number of subbits from the finally prepared cat-state. Circuits 4A, 4B, and 4C are identical except for the different number of measurements and post-selection in phase (b), which illustrates the variation in the fault-tolerant results provided by each circuit.

[0075] The “post-selection” phase (phase b) improves the quality of the states produced by the circuit. However, if the dropout rate is large (e.g., causing dropout and circuit restart when a measurement produces a significant result), the initial cat-state preparation phase (a) may be executed multiple times before success. In this case, it may be more efficient to attempt to classically correct the measurement quantum state measured in phase (b) with a nontrivial result, rather than starting preparation from scratch. Figure 5 illustrates a family of circuits that use error correction to classically correct errors in the initial cat-state instead of discarding the state and restarting at each nontrivial measurement result in phase (b). This family of circuits can therefore be understood as... Figure 3 - The "no-discard" version of the race is shown in Figure 4.

[0076] Figures 5A-5C The diagram illustrates a series of dropout-free fault-tolerant cat-state preparation circuits 502, 504, and 506, each providing fault tolerance. t And the depth is 4+4 t Similar to the above regarding... Figure 3 and Figures 4A-4C The circuits discussed, the dropout-free fault-tolerant cat preparation circuits 502, 504, and 506, each comprise three stages (a), (b), and (c). Stage (a) is the initial cat-state preparation stage, which includes the same or similar operations as those discussed above with respect to Figures 2-4. Stage (b) is the testing and classical calibration stage, including... Figure 3 - The same type of measurement (M) used in the post-selection stage (b) of Figure 4. zz However, in Figures 5A-5CIn the middle, the selected (PS) rectangle is the correction rectangle at the end of stage (b) (e.g., Figure 5A The correction rectangle 510 in step (b) is replaced by the measurement rectangle (e.g., rectangle 512). Furthermore, the measurement rectangle in step (b) represents the corresponding z-based joint parity measurement (M). zz And classical correction (N-qubit Pauli operator), which can be applied to the quantum state of one or more qubits to transform the measurement result from non-trivial to trivial.

[0077] refer to Figure 5A The correction rectangle 510 at the end of stage (b) is an N-qubit Pauli operation, which depends on all joint parity (M) operations in stage (b). zz The output of the measurement. The correction rectangle 510 receives the measurement results performed in stage (b) and the classical corrections (e.g., N-qubit Pauli operations) associated with each measurement result as input.

[0078] Figure 5A 502 Figure 5B 504 and Figure 5C Each drop-free fault-tolerant cat-state preparation circuit in the 506 is t Fault-tolerant and with a depth of 4+4 t This is because the testing phase (phase (b)) of these circuits is related to... Figure 3 The circuit shown and described in Figure 4 involves twice as many measurements as the later selection stage. This repetition of measurements within stage (b) allows for accurate determination of the classical correction (e.g., correction rectangle 510) to be applied at the end of the stage.

[0079] Proposition 2 . m-1 auxiliary qubits and depth can be used (in ) of t Fault-tolerant circuits prepare m-qubit cat states on a row of qubits This circuit is non-discardable.

[0080] Proof of Proposition 2: This proof relies on the same baseline argument as the proof of Proposition 1 above. That is, in order for the error of stage (a) of the circuit to remain undetectable throughout stage (b), there must be at least one error for every pair of consecutive measurement levels. The goal of the correction rectangle 510 is to identify the residual error caused by any misconfiguration with up to t errors. If the correction fails, it means that the residual error E of one misconfiguration ω is confused with the residual error E' of another misconfiguration ω'. Such a combined configuration ω-ω' contains at most ω-ω'<2t+1 errors, resulting in trivial results in stage (b) and nontrivial residual Pauli errors in the output state. This does not occur when 4t measurement layers are used in stage (b).

[0081] Figures 6A-6C The diagram illustrates a series of cat-state preparation circuits 602, 604, and 606, each providing fault tolerance. t And the depth is 4+2 t And related to Figures 4A-4C Compared to similar circuits disclosed elsewhere, each has a reduced circuit size. The cat-state preparation circuits 602, 604, and 606 achieve this by eliminating unnecessary parameters and then selecting measurements from... Figures 4A-4C It was obtained from the circuit. Figure 6A The 602 circuit is a fault-tolerant circuit; Figure 6B The 604 circuit is a 2-fault-tolerant circuit; and Figure 6C Circuit 606 is for all 9-qubit cat states. t t-tolerance ≥3.

[0082] It is worth noting that each measurement of the circuit introduces some additional noise. It is reasonable to expect the waiting qubits to suffer less error than the measured qubits. Figures 6A-6C and Figures 4A-4C The only difference between the series of circuits in the diagram is the removal of the post-selection measurements involving the first two qubits and the last two qubits of that row. Single errors causing these qubit errors are no longer detected. This is acceptable because a single qubit error results in a single qubit Pauli error in the output state (up to the stabilizer), which does not violate the fault tolerance condition. Extending this argument to configurations with 2 and 3 errors results in... Figure 4B and Figure 4C The circuit.

[0083] Figure 7 The illustration shows example operation 700 for preparing a cat state using a post-selection technique in a circuit with a fault tolerance level of t and a depth of 4+2t. Preparation operation 702 prepares the initial cat state for one row of qubits. Initialization operation initializes the index for the upcoming current round of measurements (m=l). In response to determining that m is still less than or equal to twice the circuit's fault tolerance level, determination operation 706 initiates the measurement round.

[0084] The selection operation 708 selects a pair of adjacent qubits from the row, and the measurement operation performs a z-basis joint parity measurement on the selected pair. The determination operation 712 evaluates the measurement result to determine if it is nontrivial. If the measurement result is nontrivial ("1"), this indicates that the prepared cat state is faulty, and the discard operation 718 discards the cat state. In this case, the preparation operation 702 prepares a new cat state, and operations 704, 706, 708, and 710 are repeated.

[0085] Assuming operation 712 determines that the measurement result is trivial (“0”), operation 714 determines whether there are any remaining adjacent qubit pairs in the current round of measurements (e.g., first round m=1, second round m=2, etc.) that have not yet undergone measurement operation 710. If adjacent qubit pairs do indeed still need to be measured, operation 708 selects new adjacent qubit pairs, and operations 710, 712, 718, etc., are repeated until all adjacent pairs have been measured in the current round “m” and all of these measurements have produced trivial results. When this condition is met, the increment operation increases the measurement round number “m”, starting a new measurement round that repeats the above operations until all adjacent qubit pairs have been measured in the second round (e.g., m=2) and all such measurements again produce trivial results. It is worth noting that... Figures 6A-6C The implementation shown does not perform a joint parity measurement on all adjacent qubit pairs, but only on a subset of these qubits (forming...). Figures 6A-6C (The triangular circuit patterning). Specifically, joint parity measurements of adjacent qubit pairs may be skipped because errors occurring at the circuit edges cannot propagate to the main body. Therefore, these errors are less harmful than errors in the main body of the circuit, and there is no need to measure these errors to detect and correct errors in the main body of the circuit.

[0086] Repeat the above process for "m" rounds of measurements, up to twice the circuit's fault tolerance level. For example, if the circuit has a fault tolerance of 1, then operation 700 performs a total of two rounds of joint parity measurements on all adjacent qubit pairs. Similarly, if the circuit has a fault tolerance of 2, then operation 700 performs a total of four rounds of joint parity measurements on all adjacent qubit pairs.

[0087] Once operation 706 determines that the number of measurement rounds performed is equal to twice the circuit's fault tolerance level, unentanglement operation 720 unentangles all even-numbered 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 those target qubits and extracting the measurement results.

[0088] Figure 8 The illustration shows an example operation 800 for preparing a cat state without dropping (e.g., without using post-selection) in a circuit with a fault tolerance level of t and a depth of 4+2t. Preparation operation 802 prepares the initial cat state for a row of qubits. Initialization operation initializes the index for the upcoming current round measurement (m=l). In response to determining that m is still less than four times the circuit's fault tolerance level, determination operation 806 starts the measurement round.

[0089] The selection operation 808 selects a pair of adjacent qubits from the row, and the measurement operation performs a z-basis joint parity measurement on the selected pair. The determination operation 812 evaluates the measurement result to determine if it is significant. If the measurement result is nontrivial (“1”), this indicates an error in the prepared cat state. In this case, the error correction operation 816 calculates and stores a classical correction (N-qubit Pauli operator), which can be used to change the quantum state of one or more stored qubits to transform the measurement result from nontrivial to trivial.

[0090] Following this (or, in all cases where the result of measurement operation 812 is trivial), determination operation 814 determines whether there are any remaining adjacent qubit pairs that have not yet been measured in the current measurement round (e.g., first round m=1, second round m=2, etc.). If adjacent qubit pairs are indeed still to be measured, selection operation 808 selects a new adjacent qubit pair, and operations 810, 812, 816, and 814 are repeated until all adjacent pairs have been measured in the current round “m” and all such measurements have produced trivial results. When this condition is met, the increment operation increases the measurement round number “m”, and a new measurement round repeating the above operations begins until all adjacent qubit pairs have been measured in the second round (e.g., m=2) and all of these measurements have again produced trivial results.

[0091] Repeating the above process for "m" rounds of measurements can achieve up to four times the circuit's fault tolerance level. For example, if the circuit has a fault tolerance of 1, then operation 800 pairs of all adjacent qubit pairs performs a total of four rounds of joint parity measurement and error correction calculations. Similarly, if the circuit has a fault tolerance of 2, operation 800 pairs of all adjacent qubit pairs performs a total of eight rounds of joint parity measurement and error correction calculations.

[0092] Once operation 806 determines that the number of measurement rounds performed is equal to four times the circuit's fault tolerance level, classical correction operation 818 corrects the classical quantum state stored for each qubit based on the calculated correction operator, and the correction calculation is stored by correction calculation 816. De-entanglement operation 820 unentangles all even-numbered 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 822 performs data measurements on two or more target qubits by connecting the resulting cat states to those target qubits and extracting the measurement results.

[0093] Figure 9An exemplary system for implementing the disclosed technology is illustrated. This system includes a general-purpose computing device in the form of an exemplary conventional PC 900, including one or more processing units 902, system memory 904, and a system bus 906 coupling various system components, including the system memory 904, to the one or more processing units 902. The system bus 906 can be any of a variety of bus architectures, including a memory bus or memory controller, a peripheral bus, and a local bus using any kind of bus architecture. The exemplary system memory 904 includes read-only memory (ROM) 908 and random access memory (RAM) 1210. A basic input / output system (BIOS) 912 contains basic routines stored in the ROM 908 that facilitate the transfer of information between components within the PC 900.

[0094] In implementation, system memory 904 stores logical operations used to prepare for cat mode, such as those generated by... Figure 7 Operation 700 and Figure 8 The operations described in 800.

[0095] The exemplary PC 900 also includes one or more storage devices 930, such as a hard disk drive for reading and writing to a hard disk, a disk drive for reading or writing to a removable disk, and an optical disc drive (e.g., as a CD-ROM or other optical medium) for reading or writing to a removable optical disc. Such storage devices can be connected to the system bus 1206 via a hard disk drive interface, a disk drive interface, and an optical disc drive interface, respectively. The drives and their associated computer-readable media provide the PC 1200 with non-volatile storage of computer-readable instructions, data structures, program modules, and other data. Other types of computer-readable media that can store data accessible by the PC (e.g., magnetic tape, flash memory cards, digital video disks, CDs, DVDs, RAM, ROM, etc.) may also be used in the exemplary operating environment.

[0096] Multiple program modules can be stored in storage device 930, which includes the operating system, one or more applications, other program modules, and program data. Decoding logic can be stored in storage device 930 and memory 904, or in addition to memory 904. Users can input commands and information into PC 900 through one or more input devices 940 (e.g., keyboard) and pointing devices (e.g., mouse). Other input devices may include digital cameras, microphones, joysticks, game controllers, satellite antennas, scanners, etc. These and other input devices are typically connected to one or more processing units 902 via a serial port interface coupled to system bus 906, but may also be connected via other interfaces, such as parallel ports, game ports, or Universal Serial Bus (USB). Monitor 946 or other types of display devices are also connected to system bus 906 via an interface such as a video adapter. Other peripheral output devices 945 may be included, such as speakers and printers (not shown).

[0097] PC 900 can operate in a network environment using logical connections to one or more remote computers (e.g., remote computer 960). In some examples, this includes one or more network or communication connections 950. Remote computer 960 can be another PC, server, router, network PC, or peer device or other public network node, and typically includes many or all of the elements described above regarding PC 900, although only memory storage device 962 is illustrated in Figure 12. PC 900 and / or remote control computer 960 can be connected to logical local area networks (LANs) and wide area networks (WANs). Such network environments are common in offices, enterprise-wide computer networks, intranets, and the Internet.

[0098] When used in a local area network (LAN) environment, PC 1200 connects to the LAN via a network interface. When used in a WAN network environment, PC 1200 typically includes a modem or other device for establishing communication over the WAN (e.g., the Internet). In a network environment, program modules or portions thereof depicted relative to PC 1200 may be stored in remote storage devices or other locations on the LAN or WAN. The network connections shown are exemplary, and other methods for establishing communication links between computers may be used.

[0099] The example method disclosed herein provides a series of operations to prepare a cat state in a quantum circuit with a fault tolerance of t and a circuit depth less than or equal to 4+4t. This series of operations includes performing a joint parity measurement sequence on at least adjacent pairs of entangled qubits to form an initial cat state; repeating the measurement sequence for at least t rounds; and untangling 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 interleaved with the first set of alternating qubits along a one-dimensional connection line, and the prepared cat state is guaranteed to have a predefined degree of determinism, including less than or equal to t One error.

[0100] According to any of the aforementioned example methods, additional operations are required to extract measurements of two or more data qubits using the prepared cat state. These additional operations involve entanglement of two or more data qubits with the prepared cat state and extracting the measurement results of two or more data qubits from the prepared cat state.

[0101] In another example method according to any of the foregoing methods, the method further includes: (1) in response to determining that any joint parity measurement produces a nontrivial result, discarding the cat state and preparing a new cat state; (2) in response to determining that no joint parity measurement produces a nontrivial result, proceeding to the next round of at least t rounds of measurement.

[0102] In yet another example method according to any of the foregoing methods, repeating the measurement sequence over at least t rounds further includes performing a 2*t round measurement sequence. 2*t rounds are sufficient to guarantee a fault tolerance of [value missing]. t The number of errors was [number], and in any case, the cat state was not discarded.

[0103] Another example of any of the foregoing methods includes: for each measurement in 2*t rounds, computing and storing an N-qubit Pauli operator that can be used to change the stored quantum state of one or more qubits to transform the measurement result from non-trivial to trivial.

[0104] In yet another example of any of the foregoing methods, the method further includes correcting a series of stored quantum states for each qubit based on computationally and stored N-qubit Pauli operators after 2*t rounds.

[0105] In yet another example of any of the foregoing methods, the method further includes preparing a ready cat state by performing an operation to entangle a series of qubits, the operation comprising: (1) performing an x-basis measurement on each of the qubits in the series; (2) for each x-basis measurement having a nontrivial result, implementing a measurement update operation to flip the result; (3) performing a z-basis joint parity measurement on each pair of adjacent qubits in the series of qubits; and for each z-basis joint parity measurement having a nontrivial result, implementing a measurement update operation to flip the result.

[0106] In yet another example of any of the aforementioned 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 in the series.

[0107] The example quantum devices disclosed in this article include those with fault tolerance. t And the circuit depth is less than or equal to 4+4 t The cat-state preparation circuit is configured to prepare a cat state by performing a sequence of joint parity measurements on individual adjacent qubit pairs in an entangled qubit group to form an initial cat state; repeating the measurement sequence for at least t rounds; and untangling a first set of alternating qubits from the initial cat state to form a prepared cat state, guaranteed to have a deterministic degree of less than or equal to t errors. The prepared cat state is formed by the remaining second set of alternating qubits, which are interleaved with the first set of alternating qubits along a one-dimensional connection line.

[0108] In another quantum device according to any of the aforementioned quantum devices, the cat state preparation circuit is further configured to perform operations to extract measurements of two or more data qubits using the prepared cat state, the operations including entanglement of two or more data qubits with the prepared cat state; and extracting the measurement results of two or more data qubits from the prepared cat state.

[0109] In another quantum device according to any of the aforementioned quantum devices, the cat state preparation circuit is also configured to discard the cat state and prepare a new cat state in response to determining that any joint parity measurement produces a nontrivial result; and to proceed to the next round of at least t rounds of measurements in response to determining that no joint parity measurement produces a nontrivial result.

[0110] In any of the aforementioned quantum devices, the cat-state preparation circuit repeats the measurement sequence for at least t rounds by executing a 2*t round measurement sequence, which is sufficient to collectively guarantee a fault tolerance of 100%. t The number of errors was zero, and the cat state was not discarded under any circumstances.

[0111] In yet another quantum device of any of the aforementioned quantum devices, the cat-state preparation circuit is also configured to compute and store an N-qubit Pauli operator for each measurement in 2*t rounds. An N-qubit Pauli operator is an operator that can be used to change the stored quantum state of one or more qubits to transform a measurement result from nontrivial to trivial.

[0112] In another quantum device of any of the aforementioned quantum devices, the cat-state preparation circuit is also configured to correct the stored quantum state of each qubit in a series based on computationally and stored N-qubit Pauli operators.

[0113] In yet another quantum device of any of the aforementioned quantum devices, the cat-state preparation circuit is further configured to prepare a cat state by performing operations to entangle a series of qubits. These operations include: performing an x-basis measurement on each of the qubits in the series; performing a measurement update operation to flip the result of each x-basis measurement to a nontrivial result; performing a z-basis joint parity measurement on each individual adjacent pair of qubits in the series; and performing a measurement update operation to flip the result of each z-basis joint parity measurement to a nontrivial result.

[0114] In any of the aforementioned quantum devices, the depth of the cat-state preparation circuit is independent of qubit connectivity, the size of the prepared cat state, and the distance between qubits in the series.

[0115] The examples disclosed herein are tangible computer-readable storage media used to store computer processes for fault-tolerant operations. t And the circuit depth is less than or equal to 4+4 t The processor-executable instructions for preparing a cat state in a quantum circuit are described. The computer process includes: performing a joint parity measurement sequence on individual adjacent qubit pairs in entangled qubits to form an initial cat state that repeats the measurement sequence over at least t rounds; and untangling a first set of alternating qubits from the initial cat state to form a prepared cat state consisting of a second set of alternating qubits. The second set of alternating qubits is interleaved with the first set of alternating qubits along a one-dimensional connection line. The prepared cat state is guaranteed to have a predefined degree of determinism to include less than t One error.

[0116] In another example of a computer-readable storage medium according to any of the foregoing storage media, the computer process further includes: in response to determining that any joint parity measurement produces a nontrivial result, discarding the cat state and preparing a new cat state; and in response to determining that no joint parity measurement produces a nontrivial result, proceeding to the next round of at least t rounds of measurement.

[0117] In another example of a computer-readable storage medium according to any of the foregoing storage media, the computer process further includes: performing a 2*t round measurement sequence. 2*t rounds are sufficient to guarantee a fault tolerance of up to t error numbers with a predefined degree of determinism, and in no case is the cat state discarded.

[0118] In another example of a computer-readable storage medium according to any of the foregoing storage media, the computer process further includes: for each measurement of 2*t rounds, calculating and storing an N-qubit Pauli operator that can be used to change the quantum state of one or more stored qubits to transform the result of the measurement from nontrivial to trivial.

[0119] The example system d disclosed in this paper includes a series of operations to perform fault tolerance. t And the circuit depth is less than or equal to 4+4 t A device for preparing a cat state in a quantum circuit. The system includes at least means for performing a joint parity measurement sequence on adjacent pairs of entangled qubits in a series of qubits to form an initial cat state; means for repeating the measurement sequence for at least t rounds; and means for untangling 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 interleaved with the first set of alternating qubits along a one-dimensional connection line, and the prepared cat state is guaranteed to have a predefined degree of determinism, including less than or equal to t One error.

[0120] The foregoing specification, examples, and data provide a complete description of the structure and use of exemplary implementations. Since many implementations can be made without departing from the spirit and scope of the claimed invention, the appended claims define the invention. Furthermore, structural features of different examples can be combined in yet another implementation without departing from the cited claims.

Claims

1. A method for preparing a cat state, comprising: A series of operations are performed in a quantum circuit with a fault tolerance t and a circuit depth of less than or equal to 4+4t to prepare a cat state, said series of operations including at least: Perform a joint parity measurement sequence on individual adjacent qubit pairs in an entangled series of qubits to form an initial cat state; Repeat the measurement sequence for at least t rounds; and The first set of alternating qubits is unentangled from the initial cat state to form a prepared cat state with a remaining second set of alternating qubits, which are interleaved with the first set of alternating qubits along a line of one-dimensional connectivity. The prepared cat state is guaranteed to have a predefined degree of determinism to include an error number less than or equal to t.

2. The method according to claim 1, further comprising: The prepared cat state is used to perform additional operations to extract measurements of two or more data qubits, the additional operations including: Entangling the two or more data qubits with the prepared cat state; and The measurement results of the two or more data qubits are extracted from the prepared cat state.

3. The method according to claim 1, further comprising: In response to the determination that any of the joint parity measurements produces a nontrivial result, the cat state is discarded and a new cat state is prepared. as well as In response to the determination that no joint parity measurement produces a nontrivial result, proceed to the next round of the at least t rounds of measurement.

4. The method of claim 1, wherein repeating the measurement sequence over at least t rounds further comprises: The measurement sequence is executed in 2*t rounds, which is sufficient to guarantee a fault tolerance of up to t number of errors with the predefined degree of determinism, and the cat state is not discarded under any circumstances.

5. The method according to claim 4, further comprising: For each measurement in the 2*t rounds, an N-qubit Pauli operator is calculated and stored. The N-qubit Pauli operator can be used to change the stored quantum state of one or more of the qubits to transform the measurement result from non-trivial to trivial, where N is the number of qubits in the initial cat state.

6. The method according to claim 5, further comprising: After the 2*t rounds, the stored quantum state of each of the qubits in the series is corrected based on the calculated and stored N-qubit Pauli operator.

7. The method according to claim 1, further comprising: The prepared cat state is prepared by performing an operation sequence to entangle the series of qubits, the operation sequence including: Perform an x-basis measurement on each of the qubits in the series; For each of the x-basis measurements that has a nontrivial result, implement a measurement update operation to flip the result; For each individual adjacent qubit pair in the series of qubits, perform a z-basis joint parity measurement; and For each of the z-based joint parity measurements with nontrivial results, a measurement update operation is implemented to flip the results.

8. The method of claim 1, wherein 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 in the series.

9. A quantum device, comprising: A cat-state preparation circuit, having a fault tolerance t and a circuit depth less than or equal to 4+4t, is configured as follows: Perform a joint parity measurement sequence on individual adjacent qubit pairs in an entangled series of qubits to form an initial cat state; Repeat the measurement sequence for at least t rounds; and The first set of alternating qubits is unentangled from the initial cat state to form a prepared cat state, which is guaranteed to have a predefined degree of determinism to include an error number less than or equal to t. The prepared cat state is formed by the remaining second set of alternating qubits, which are interleaved with the first set of alternating qubits along a line of one-dimensional connectivity.

10. The quantum device of claim 9, wherein the cat-state preparation circuit is further configured to: The prepared cat state is used to perform operations to extract measurements of two or more data qubits, the operations including: Entangling the two or more data qubits with the prepared cat state; and The measurement results of the two or more data qubits are extracted from the prepared cat state.

11. The quantum device of claim 9, wherein the cat-state preparation circuit is further configured to: In response to determining that any of the joint parity measurements produces a nontrivial result, the cat state is discarded and a new cat state is prepared; and In response to the determination that no joint parity measurement produces a nontrivial result, the next round of at least t rounds of measurement continues.

12. The quantum device of claim 9, wherein the cat-state preparation circuit repeats the measurement sequence in the at least t rounds of measurement in such a manner as follows: The measurement sequence is executed in 2*t rounds, which is sufficient to guarantee a fault tolerance of up to t errors with the predefined degree of determinism, and the cat state is not discarded under any circumstances.

13. The quantum device of claim 12, wherein the cat-state preparation circuit is further configured to: For each measurement in the 2*t rounds, an N-qubit Pauli operator is calculated and stored. The N-qubit Pauli operator can be used to change the stored quantum state of one or more of the qubits to transform the result of the measurement from non-trivial to trivial, where N is the number of qubits in the initial cat state.

14. The quantum device of claim 13, wherein the cat-state preparation circuit is further configured to: The stored quantum state of each of the qubits is corrected based on the computation and storage of the N-qubit Pauli operator.

15. The quantum device of claim 9, wherein the cat-state preparation circuit is further configured to: The prepared cat state is prepared by entangled qubits through a sequence of operations, the sequence of operations including: Perform an x-basis measurement on each of the qubits in the series; For each x-based measurement with a nontrivial result, implement a measurement update operation to flip the result; Perform a z-basis joint parity measurement on each individual adjacent pair of qubits in the series of qubits; as well as For each of the z-based joint parity measurements with nontrivial results, a measurement update operation is implemented to flip the results.

16. The quantum device of claim 9, wherein the depth of the cat-state preparation circuit is independent of qubit connectivity, the size of the prepared cat state, and the distance between the qubits in the series.

17. One or more tangible computer-readable storage media storing processor-executable instructions for performing a computer process in a quantum circuit having a fault-tolerant t and a circuit depth of less than or equal to 4+4t to prepare a cat state, the computer process comprising: Perform a joint parity measurement sequence on individual adjacent qubit pairs in an entangled series of qubits to form an initial cat state; Repeat the measurement sequence for at least t rounds; and The first set of alternating qubits is unentangled from the initial cat state to form a prepared cat state with a remaining second set of alternating qubits, which are interleaved with the first set of alternating qubits along a line of one-dimensional connectivity. The prepared cat state is guaranteed to have a predefined degree of determinism to include an error number less than or equal to t.

18. One or more tangible computer-readable storage media according to claim 17, wherein the computer process further comprises: In response to the determination that any of the joint parity measurements produces a nontrivial result, the cat state is discarded and a new cat state is prepared. as well as In response to the determination that no joint parity measurement produces a nontrivial result, proceed to the next round of the at least t rounds of measurement.

19. One or more tangible computer-readable storage media according to claim 17, wherein the computer process further comprises: The measurement sequence is executed in 2*t rounds, which is sufficient to guarantee a predefined degree of determinism with a fault tolerance of t errors, and the cat state is not discarded under any circumstances.

20. The one or more tangible computer-readable storage media of claim 19, wherein the computer process further comprises: For each measurement in the 2*t rounds, an N-qubit Pauli operator is calculated and stored. The N-qubit Pauli operator can be used to change the stored quantum state of one or more of the qubits to transform the result of the measurement from non-trivial to trivial, where N is the number of qubits in the initial cat state.

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