Magic state distillation using internal and external error correction codes

Abstract

Disclosed examples relate to extracting a magic state on a quantum computing device, the magic state encoded in a [[n, k, d]] block code including an outer code. One example provides a method that includes preparing an encoded noisy magic state using data qubits and measuring a Clifford stabilizer on the data qubits, thereby applying an inner code. The method further includes initializing output qubits and initiating a teleportation of an extracted magic state derived from the encoded noisy magic state to the output qubits. The method further includes measuring an X stabilizer on the data qubits, post-selecting based on the result, destructively measuring each data qubit with a Z stabilizer, and applying one or more post-selection conditions to the data qubits to complete the teleportation of the extracted magic state to the output qubits.

Description

Background Technology

[0001] Quantum computers use the superposition and entanglement of quantum states to perform quantum computation. Quantum states provide observable probability distributions. A qubit stores information about the probability distribution of a single state. Quantum computation is performed using qubits and applications of logic operations on those qubits. Universal quantum computing can be achieved by combining high-precision Clifford's operations with quantum states known as magic states. Summary of the Invention

[0002] This summary provides a simplified overview of the options for concepts further described in the detailed description below. This summary is not intended to identify key or essential features of the claimed subject matter, nor is it intended to limit the scope of the claimed subject matter. Furthermore, the claimed subject matter is not limited to implementations that address any or all of the shortcomings mentioned in any part of this summary.

[0003] The disclosed examples relate to extracting a magic state on a quantum computing device, the magic state being encoded in a [[n,k,d]] block code including an outer code. One example provides a method comprising preparing an encoded noisy magic state using data qubits, and measuring a Clifford stabilizer on the data qubits to apply an inner code. The method further comprises initializing an output qubit and initiating teleportation of the extracted magic state derived from the encoded noisy magic state to the output qubit. The method further comprises measuring an X stabilizer on the data qubits, performing post-selection based on the result, destructively measuring each data qubit using a Z stabilizer, and applying one or more post-selection conditions to the data qubits to complete the teleportation of the extracted magic state to the output qubit. Attached Figure Description

[0004] Figure 1 An example quantum computer is illustrated schematically.

[0005] Figure 2 The illustration shows the Bloch sphere, which graphically represents the quantum state of a quantum bit in a quantum computer.

[0006] Figure 3 An example measurement sequence is schematically shown for teleporting arbitrary states to qubits and for injecting T-states or S-states by measurement.

[0007] Figure 4 An example scheme for teleporting magic-state stealth to block code under given geometric constraints is illustrated.

[0008] Figure 5 An example layout of the [[n,k,d]] block code is shown schematically.

[0009] Figure 6 A flowchart is shown for an example method used to extract the magic state.

[0010] Figure 7 A block diagram of an example computing system is shown. Detailed Implementation

[0011] Many schemes for quantum computing rely on first implementing a set of high-precision operations known as Clifford operations. To achieve universal quantum computing, it is necessary to implement additional operations, such as T-gates or CCZ gates. Inevitably, both Clifford operations and T-gates introduce some error. Large-scale, fault-tolerant quantum computers rely on various error correction codes. Essentially, to achieve fault tolerance, error correction must be faster than their generation. In the case of universal quantum computing via Clifford operations, two types of error correction are performed. Error correction codes are used to construct Clifford gates, and additional error correction codes are used to generate high-precision T-gates.

[0012] Error correction is performed on quantum states that have been redundantly encoded. Instead of measuring a single qubit, parity checks are performed across multiple qubits to detect X-errors, bit-flip errors, Z-errors, or phase-flip errors. Error reduction in Clifford operations can be performed using quantum error correction codes of the stabilizer type, referred to in this paper as "inner codes." To reduce errors in T-gates, assuming the Clifford operation is highly error-free, a method called magic state extraction can be used, which also employs error correction codes, referred to in this paper as "outer codes." Combined, these error correction codes impose a significant overhead on quantum computers.

[0013] Therefore, examples of magic state extraction using the outermost code as a fully quantum error correction code are disclosed. The extraction code can be used to stabilize Clifford gates, thereby reducing the necessary overhead of Clifford gates. Example extraction protocols are disclosed for extracting noisy T states into high-fidelity T states or CCZ magic states. In some examples, the protocol is a custom implementation of a known abstract error correction code.

[0014] In some examples, the extraction protocol is based on the idea of ​​measuring the Clifford (rather than Pauli) stabilizer of the T state (see, for example, E. Knill, 2004, arXiv.quant-ph / 0402171v1; E. Knill, 2004arXiv:quant-ph / 0404104v1, hereinafter Knill; P. Aliferis, D. Gottesmann, J. Preskill, Quant. Inf. Comput. 6, 97-165, 2006, arXiv:quant-ph / 0504218; and J. Haah, M. Hastings, D. Poulin, D. Wecker, Quantum 1, 31, 2017, arXiv:1703.07847v1). In other examples of the CCZ extraction protocol, the protocol can be generalized using triorthogonal coding (ET Campbell and M. Howard, Phys. Rev. A 95, 022316, 2017, arXiv:1209.2426, hereinafter referred to as Campbell) (see, for example, S. Bravyi and J. Haah, “Magic-state distillation with low overhead,” Phys. Rev. A 86, 052329, 2012, hereinafter referred to as Bravyi).

[0015] Before discussing magic state extraction, let's first discuss some aspects of quantum computer architecture. Figure 1 Some aspects of an example quantum computer 10 configured to perform quantum logic operations are shown. The quantum computer 10 includes at least one qubit register 12 comprising an array of qubits 14. The illustrated qubit register is 8 qubits long. It will be understood that other example quantum computers may include longer and shorter qubit registers, and may include two or more qubit registers of any suitable length.

[0016] Depending on the desired structure of the quantum computer 10, the qubits 14 of the qubit register 12 can take various forms. As examples, each qubit may include a superconducting Josephson junction, trapped ions, trapped atoms coupled to a high-fineness cavity, atoms or molecules confined within a fullerene, ions or neutrally doped atoms confined within a main lattice, quantum dots exhibiting discrete spatial or spin-electronic states, electron-hole pairs in a semiconductor junction trapped by electrostatic traps, coupled quantum wire pairs, magnetically resonant addressable atomic nuclei, Majorana fermions, free electrons in helium (He), molecular magnets, or metalloid carbon nanospheres. Generally, each qubit 14 may include any particle or any particle system that can exist in two or more discrete quantum states that can be experimentally measured and manipulated. For example, qubits can be realized in multiple processing states corresponding to different modes of light propagation through linear optical elements (such as mirrors, beam splitters, and phase shifters), or in states accumulated within a Bose-Einstein condensate. The quantum computer 10 may include suitable hardware to carry qubits 14. For example, Majorana lines can carry Majorana fermions.

[0017] The quantum state of a qubit can be represented on a Bloch sphere 200, such as... Figure 2 The points on the surface of the Bloch sphere comprise all possible pure states |ψ> of the qubit, while the points inside correspond to all possible mixed states. The north and south poles of the Bloch sphere 200 correspond to the standard basis vectors |0> and |1>, respectively. For example, these basis vectors can represent the spin-up and spin-down states of electrons or other fermions. A mixed state is any state that can be described as a mixture of pure states. Mixed states may be caused by decoherence, which may occur due to poor coupling with external degrees of freedom. Quantum operations implemented via the quantum logic gates described herein are equivalent to rotations or a series of rotations around the axis of the Bloch sphere. For example, the Pauli-X gate is equivalent to a rotation of p radians around the X-axis of the Bloch sphere.

[0018] The quantum states and operations discussed in this article include:

[0019] X = |1><0| + |0><1|

[0020] Z = |0><0|-|1><1|

[0021] T = |0><0| + e iπ / 4 |1><1|

[0022] S = T 2=|0><0|+i|1><1|

[0023] |CCZ>=∑ a,b,c＝0,1 (-1) abc |a,b,c>and

[0024] |±>=0>±1>.

[0025] Return to Figure 1 The quantum computer 10 includes a controller 18. This controller may include at least one processor 20 and associated computer memory 22. The processor 20 of the controller 18 may be operatively coupled to peripheral components, such as network components, to enable remote operation of the quantum computer. The processor 20 of the controller 18 may take the form of a central processing unit (CPU), a graphics processing unit (GPU), or a similar component. Therefore, the controller may include conventional electronic components. The terms “conventional” and “non-quantum” herein apply to any component that can be precisely modeled as a collection of particles, without regard to the quantum state of any individual particle. For example, classical electronic components include integrated transistors, resistors, and capacitors. The computer memory 22 may be configured to store program instructions 24 that cause the processor 20 to perform any function or process of the controller. In an example where the qubit register 12 is a cryogenic or low-temperature device, the controller 18 may include control components that can operate at cryogenic or low-temperature conditions—for example, a field-programmable gate array (FPGA) operating at 77 Kelvin. In such an example, the cryogenic control components are operatively coupled to interface components that can operate at room temperature.

[0026] The controller 18 of the quantum computer 10 is configured to receive multiple inputs 26 and provide multiple outputs 28. The inputs and outputs may each include digital and / or analog lines. At least some of the inputs and outputs may be data lines through which data is provided to and / or extracted from the quantum computer. Other inputs may include control lines through which the operation of the quantum computer can be adjusted or otherwise controlled.

[0027] The controller 18 is operatively coupled to the qubit register 12 via a quantum interface 30. The quantum interface is configured to exchange data bidirectionally with the controller. The quantum interface is also configured to exchange signals corresponding to the data bidirectionally with the qubit register. Depending on the architecture of the quantum computer 10, such signals may include electrical, magnetic, and / or optical signals. The signals teleported via the quantum interface allow the controller to query and otherwise influence the quantum states stored in the qubit register, such as those defined by the collective quantum states of the array of qubits 14. For this purpose, the quantum interface includes at least one modulator 32 and at least one demodulator 34, each operatively coupled to one or more qubits of the qubit register. Each modulator is configured to output a signal to the qubit register based on modulated data received from the controller. Each demodulator is configured to sense the signal from the qubit register and output data to the controller based on that signal. In some examples, the data received from the demodulator may be an observable estimate of a measurement of the quantum state held in the qubit register.

[0028] In some examples, a properly configured signal from modulator 32 can physically interact with one or more qubits 14 of qubit register 12 to trigger a measurement of the quantum state held in one or more qubits. Demodulator 34 can then sense the resulting signal emitted by one or more qubits according to the measurement and can provide data corresponding to the resulting signal to controller 18. In other words, the demodulator can be configured to output an estimate of one or more observed variables reflecting the quantum state of one or more qubits of the qubit register based on the received signal and provide that estimate to the controller. In a non-limiting example, the modulator can provide appropriate voltage pulses or pulse sequences to the electrodes of one or more qubits based on data from the controller to initiate the measurement. The demodulator can sense photon emission from one or more qubits and can assert the corresponding digital voltage level on the quantum interface line to the controller. Generally, any measurement of a quantum mechanical state is defined by an operator O corresponding to the observable to be measured; the result R of the measurement is guaranteed to be one of the allowed eigenvalues ​​of O. In quantum computer 10, R is statistically related to the qubit-register state before the measurement, but is not uniquely determined by the qubit-register state.

[0029] Based on appropriate inputs from controller 18, quantum interface 30 can be configured to implement one or more quantum logic gates to operate on quantum states stored in qubit register 12. While the function of each type of logic gate in a conventional computer system is described by a corresponding truth table, the function of each type of quantum gate is described by a corresponding operator matrix. The operator matrix operates (i.e., multiplies) on the complex vector representing the qubit register state and affects a specified rotation of that vector in Hilbert space.

[0030] The list of quantum gates above is not exhaustive and is provided for illustrative purposes only. Other gates may include Pauli-X gates, Pauli-Y gates, Pauli-Z gates, Toveley gates, Ising gates, Deutsch gates, or Hadamard gates as non-limiting examples. Some example protocols disclosed herein follow scenarios where there are geometric constraints and limitations on the basic gates. In the examples described below, the basic operations are horizontal ZZ and vertical XX measurements on nearest-neighbor qubits on a square qubit lattice, and X and Z measurements on a single qubit. In some examples, Hadamard gates are not used.

[0031] As mentioned above, universal quantum computers based on Clifford gates must be augmented with additional, non-Clifford operations. The states that perform these operations are called magic states; two examples are the single-qubit T state and the three-qubit CCZ state. To sufficiently reduce errors, the quality of the magic state should be similar to that of the Clifford gate. As described above, magic state extraction methods extract many low-quality or noisy magic states into a smaller number of high-quality magic states. The set of qubits and gates responsible for performing the extraction in a quantum computer can be called a magic state factory.

[0032] In the disclosed examples, the example protocol for magic state extraction involves a two-level error correction code: the inner code and the outer code mentioned above. The extraction protocol can use either the full error correction code or a partial error correction code as the outermost code. The outer code can be used to correct for X and Z errors and can further help avoid or correct associated errors. The example method further utilizes the error correction properties of the inner code. In some examples, the error correction method can still function despite limitations such as finite connectivity.

[0033] Logical qubits containing quantum states can be encoded across multiple physical qubits, a process known as surface code patching or "patching." Qubits encoded in larger patches may be better protected (better error correction) than those encoded in smaller patches, and higher-quality qubits may require even larger surface code patches. The example methods disclosed in this paper generate extracted magic states on independent output qubits, which may be surface code patches. In some examples, the method involves embedding the qubit into a higher-quality qubit, which is equivalent to increasing the patch size. In some examples, this embedding occurs on a single qubit throughout the extraction protocol.

[0034] As mentioned above, in some examples, there may be strict restrictions on the basic gates. In addition to the output qubits discussed above, each operation of each example protocol disclosed herein involves a multi-qubit Pauli operator over some integer m. Measurements can be performed on single qubits in X-based and Z-based systems, as well as on single qubit units composed of X, Z, and T. Quantum states can be teleported from one qubit to another via measurement. Figure 3 Two such schemes, 300 and 302, are described for teleportation to arbitrary states using ZZ or XX measurements and single-qubit measurements, respectively. Figure 3 As described in scheme 304, T-states can be injected via ZZ and X measurements, i.e., they can be stealthily transmitted into the block code. Similarly, S-states can also be injected via scheme 306.

[0035] However, in some examples, when geometric constraints only allow horizontal ZZ measurements and vertical XX measurements on a square grid of qubits, another injection scheme can be used for the T state. For example... Figure 4 As shown, scheme 400 can be used to inject T-states under these constraints. However, depending on the ZZ measurement results of -1, S-correction may be required, in which case scheme 400 can be followed by scheme 420.

[0036] Multi-qubit measurements can be performed using cat states. In some examples, an n-qubit cat state can be generated by following a process using 2n-1 qubits and single-qubit X and nearest-neighbor ZZ measurements. Using cat states, for example... The parity of the cat-state qubit and the data qubit can be measured by taking the perpendicular XX measurement between them. These measurements can then be followed by a single-qubit Z measurement on the cat-state qubit and a subsequent Pauli correction on the data qubit to bring the data qubit back to the correct post-measurement state.

[0037] An example layout of the qubits used in the protocol is shown in Figure 5The text describes an example of [[n,k,d,]] block code comprising an array of qubits. Figure 5 The qubits represented in the block can be surface code patches instead of single physical qubits. In layout 500, the first row 504 and the second row 506 are occupied by S-states and T-states, respectively. These S-states and T-states are to be consumed; layout 500 shows the qubits before the T-states are consumed. Every other qubit in the third row 508 is occupied by the data qubits of the block code. The fourth row 510 contains qubits for cat-state preparation. During the extraction protocol, noisy T-states can be placed on one side of the qubit arrays of qubits 501 and 502. In this example, the [[n,k,d]] stable subcode operates in the third row 508. The empty qubits in the third row, such as qubit 514, can be used to inject T-states via horizontal ZZ measurements.

[0038] In some examples, there will be k+2 high-quality output qubits, two of which are byproducts. Therefore, Figure 5 Layout 520 in the diagram can represent the case when k=3 and five larger output patches 522, 523, 524, 525, and 526 are generated. The patches can be dynamically changed; for example, the size of patch 524 might be reduced. After the S-states and T-states are consumed, the output qubit can interact with the qubits encoded by the block code. As mentioned above, the output qubit may have a better quality and occupy a larger surface code patch compared to other qubits. In some examples, to reduce quadratic errors, the size of the surface code patch used for the better-quality qubit might be approximately twice that of other qubits. In other examples, the precise size can be determined based on the desired quality of the output qubit. In some examples, the space occupied by the S-states and T-states, i.e., rows 504 and 506, can be used for the output qubit. For example, patch 524 could occupy the space previously occupied by the S-states and T-states. Furthermore, before the T state is consumed, it contains qubits 501 and 502, which are in the leftmost column of layout 500. After the T state in layout 520 is consumed, it may be occupied by output qubits 523 and 526.

[0039] The error model will now be discussed. Each measurement is flipped with a probability p∈[0,1). After any operation, including the identification, each qubit is affected by independent noise. The error of a single qubit is simulated by a quantum channel.

[0040] ε(ρ)=(1-p t )ρ+pD(ρ) (1)

[0041] Where D is another quantum channel. Note that the measurement error and single-qubit noise use the same p. Any T state and any T gate or Behind the door were...

[0042] ε(ρ)=(1-p t )ρ+p t D(ρ) (2)

[0043] But not through ε. Note that the same D is used in formulas (1) and (2) for simplicity.

[0044] The output qubits can be considered noise-free. Using a family of surface codes, it might be sufficient to simply choose a code distance that matches the quality of the extracted magic state. One might think that, in the context of... After measurement with this multi-qubit operator, how is the independent noise assumption maintained? The native preparation of the cat state introduces correlated errors, and measurements using such a cat state may invalidate the independent noise assumption. Therefore, a more refined protocol can be used to prepare the cat state, which differs from the ideal cat state caused by independent noise on the constituent qubits. The protocol above achieves this goal.

[0045] As mentioned above, in some examples, the magic state encoded in the block code is teleported to the output qubit in the disclosed example. This can be achieved, for example, by preparing a Z... out = +1 state and measurement X out X and Z are used for execution, where X and Z are logical operators of the block code. Using surface code patches, the output qubits can be initialized and Z measured on the code. However, for X... out The measurement of X is unusual because the data qubits of the block code have a small size. Here, a cat state can be prepared and used for measurement. An example protocol for teleporting a state to a large surface code patch via the cat state is as follows. First, let a and b represent two large surface code patches, such as patches 525 and 526. Figure 5 As indicated by arrow 530, patch b can dynamically change size. Then, prepare the cat state as follows.

[0046] 1) Preparation |00> ab And using Pauli calibration to measure X a X b To obtain |00> ab +|11> ab ,

[0047] 2) Reduce patch b to match the patch size in the fourth row of the 4×(2n-1) rectangle, and

[0048] 3) Measure the ZZ of the nearest neighbors in the fourth row and discard every other qubit.

[0049] Step 3 above can be implemented as described above. Since the leftmost pairwise measurements are performed on the large patch, the output qubit may be protected. Once the cat state is protected, X... out The measurement of X can be completed through the following procedure:

[0050]

[0051] Next, we will discuss an example protocol for extracting magic states from noisy T states.

[0052] In various examples, the magic state extraction protocol extracts the magic state, which is encoded in a [[n,k,d]] block code that includes the outcode. This [[n,k,d]] block code can be similar to... Figure 5 The layout depicted, or any other suitable layout, is arranged. First, k encoded magic states are prepared using n data qubits. Figure 5 In a specific example, the positions of the n data qubits are represented by dashed circles 508 and may include surface code patches. Each of the n data qubits may be a logical qubit with some further internal code. The method further includes applying the internal code to measure the Clifton stabilizer on the n data qubits. The method further includes initializing k output qubits and then initiating teleportation of the k output qubits to k extracted magic states. These k extracted magic states are derived from k encoded noisy magic states.

[0053] The method further includes measuring the X stabilizer on the n data qubits and performing post-selection on all +1 results. Here, the X stabilizer can be a specific stabilizer configured for the [[n,k,d]] block code. Post-selection on +1 results means that if the measurement produces a +1 result, the output qubit can be considered to have sufficient mass and can be used in quantum computing. If the measurement produces different results, the error is too large and the result may be discarded, meaning the mass of the output qubit is insufficient for quantum computing. In some examples, the method can be terminated and restarted on unfavorable post-selection results. The method further includes destructively measuring each data qubit using the Z stabilizer and applying one or more post-selection conditions to the n data qubits to complete the teleportation of k extracted magic states to k output qubits. If all post-selections are successful, the output qubits are used for quantum computing.

[0054] In some examples, the extraction method produces k+2 output qubits, comprising k magical states and 2 appendages. The output qubits may include, for example... Figure 5The surface code patch depicted in blocks 522-526.

[0055] In a more specific example protocol, the magic-state extraction method includes an outer code, which is a second-order normal weak self-dual CSS code represented by [[6,2,2]]. Here, n=6, k=2, and d=2. The steps of the protocol are as follows.

[0056] 1) In Initialize 6 data qubits and... and It is placed on one side of the qubit array.

[0057] 2) Measure XXXXII and IIXXXX on 6 data qubits and apply Z measurement to 6 data qubits to obtain logic qubits including logic states with ZIZIZI≈IZIZIZ≈+1.

[0058] 3) By measuring X t1 X1, X t2 X2, then measure Z t1 Z t2 Then, Pauli Z-correction is performed on the logical qubits to teleport the states of qubits t1 and t2 to the logical qubits. Pauli Z-correction is based on X... t1 X1, X t2 X2 and Z t1 Z t2 The measurements are X1 and X2, which are logical operators: X1 = XIXIXI, X2 = IXIXIX.

[0059] 4) Applications in data qubits

[0060] 5) Measurement in data qubits Twice, and then select the next result for all +1 results.

[0061] 6) Applications in data qubits

[0062] 7) In Initialize two output qubits o1 and o2.

[0063] 8) Measure X o1 X1 twice, the result is x1,x′1.

[0064] Measure X o2 X2 twice, the result is x2,x′2.

[0065] For the consistent results x1=x′1 and x2=x′2, perform a follow-up selection, and

[0066] If x1 = -1, apply Z o1 If x2 = -1, apply Z. o2 .

[0067] 9) Measure XXXXII and IIXXXX on 6 data qubits, and

[0068] Choose the next result from all results that are +1.

[0069] 10) Perform destructive measurements on individual data qubits on the Z-basis to obtain results z1,...z6, and perform post-selection in two cases.

[0070] z1z2z3z4=+1 and z3z4z5z6=+1 and

[0071] If z1z3z5 = -1, apply X o1 If z2z4z6 = -1, apply X. o2 .

[0072] Here, steps 1, 2, and 3 complete the preparation of two encoded noise T states for the six data qubits. In steps 4, 5, and 6, the Clifford stabilizer is measured on the six data qubits. The two output qubits are initialized in step 7. In step 8, the two encoded magic states are teleported to the two output qubits. Step 9 involves measuring the X stabilizer on the six data qubits and performing post-selection based on the results. Step 10 involves a destructive measurement on each data qubit using the Z stabilizer and applying one or more post-selection conditions to the n data qubits to complete the teleportation of k extracted magic states to the k output qubits. The two output qubits comprise two extracted T states.

[0073] The counting method for measuring depth is as follows. In steps 1, 3, and 10, m1 = 3 rounds of single-qubit measurements are performed on the data qubit. In steps 2, 3, 5, and 9, m2 = 2 + 2 + 2 + 2 = 8 rounds of multi-qubit X measurements are performed on the data qubit. In steps 4 and 6, there are 2 rounds of T-gate measurements on the data qubit, which will involve the input T-state and S-state. The injection of the T-gate requiring S-correction involves 6 single-qubit and 6 two-qubit measurements (e.g., see...). Figure 4 During other steps, the T state can be teleported immediately adjacent to the data qubit, while the S state lies above the T state. Therefore, m t =12 rounds of single-qubit or two-qubit measurement. In step 3, there is an m-value between the output qubit and the data qubit. out =4 rounds of joint measurement. The initialization of the output qubits can be neglected because it can be done in parallel with the previous data qubit measurements. In total, there are m1+m t=15 rounds of single-qubit and double-single-qubit measurements, m2 = 8 rounds of multi-qubit X measurements on the data qubit, m out =4 rounds involving X measurement of the output qubits.

[0074] To illustrate the measurement count using surface code patches more specifically, consider a synthesis measurement of a surface code patch as a unit of time. Assume that for a logic operation, d rounds of synthesis measurements are required for the surface code patch of the data qubit, and d′ rounds of synthesis measurements are required for the output qubit; d or d′ is the coding distance of the patch. To prepare a “2-fold fault-tolerant” cat state, 8 rounds of logic operations are required across the nearest patch. This time is long enough that all explicit one-qubit and two-qubit measurements in steps 1, 4, and 6 can be ignored, as they can be run in parallel with the cat state preparation in the next step. Step 9 can also be ignored.

[14] The final round of cat state preparation includes single-qubit X measurements for qubits that do not participate in the final cat state, which can be run in parallel with XX measurements between the cat state qubit and the data qubit. Including single-qubit Z measurements, it is concluded that a multi-qubit X measurement takes 9d time, and if the output qubit is involved, it takes 7d+2d′ time. Therefore, the total time is m2·9d+m out (7d+2d′). The number of physical qubits used is 4.12d. 2 +4d′ 2 It neglects the auxiliary components used for comprehensive measurement of surface codes.

[0075] In another example protocol, the magic-state extraction method includes an outer code, which is a third-order normal weak self-dual CSS code represented by [[7,1,3]]. Here, n=7, k=1, and d=3. The steps of the protocol are as follows:

[0076] 1) In Initialize 7 data qubits and... Placed on one side of the qubit array,

[0077] 2) Measure three X-stable subatomic qubits XIXIXIX, IXXIIXX, and IIIXXXX on 7 data qubits, and apply Z-measurement on 6 data qubits to obtain logical qubits including the logical state ZZZZZZZ≈+1.

[0078] 3) By measuring X t X and then Z are measured. t Invisibly transmitted into the code, the measurement X is a logical operator.

[0079] 4) Measure the three X stabilizers and perform a post-selection on all +1 results;

[0080] 5) Applications in data qubits

[0081] 6) Measuring on data qubits Twice, and then select the next result from all +1 results, and

[0082] 7) Applications in data qubits

[0083] 8) Initialize the output qubit in the state |0>_o;

[0084] 9) Measure the three equivalent X logic operators X multiplied by the X operator on the output qubit. o (IXIXIXI), X o (XIIXXII), X o (XXXIIII), the result is x1,x2,x3=±1. A subsequent selection is made based on the consistent result x=x1=x2=x3, and...

[0085] If x = -1, then apply Z. o ;

[0086] 10) Measure the three X stabilizers and perform a post-selection on all +1 results; and

[0087] 11) Destructively measure all data qubits on the Z-basis to obtain the results z1,...z7, and then perform a post-selection on three conditions: z1z3z5z7=+1, z2z3z6z7=+1, and z4z5z6z7=+1.

[0088] If z1z3z5 = -1, then apply X. o .

[0089] Here, steps 1, 2, and 3 complete the preparation of the encoded noise T-state in the 7 data qubits. Step 4 checks the X stabilizer. Steps 5, 6, and 7 measure the Clifford stabilizer of the encoded T-state. Clifford stabilizers are composed of This is caused by [the following]. Step 4 is not similar to the previous protocol disclosed above. The error check in Step 4 avoids two error processes, namely, one Z error in Steps 1, 2, and 3 and another error in Step 7, which can cancel each other out, allowing an incorrect T state to pass the Clifford stabilizer check. Steps 8, 9, 10, and 11 combine the teleportation of the encoded T state with the Pauli stabilizer check. The two output qubits are initialized in Step 8. In Step 9, the teleportation of the two encoded magic states to the two output qubits is initiated. Step 9 uses the representation of three logic operators to detect second-order processes that may lead to teleportation errors. Step 10 involves measuring the X stabilizer on the 6 data qubits and performing a post-selection based on the result. In Step 10, the data qubits are destructively measured using the Z stabilizer to complete the teleportation. The output qubits include an extracted T state.

[0090] The counting method for the measurement depth is as follows. As mentioned before, only the number of multi-qubit measurements is considered. Here, a "3-fold fault-tolerant" cat state may be required, which may require 10 rounds of single-qubit and two-qubit measurements. For those not involving output qubits, there are 3+3+2+3=11 measurements in steps 2, 4, 6, and 10. For those involving output qubits, there are 1+3=4 measurements in steps 3 and 9. Using surface code patches with distances of d (data) and d′ (output), there are 11·11d+4·(9d+2d′) rounds of comprehensive measurements for the surface code patches. The number of qubits used is 56d. 2 +3d′ 2 The appendages of the surface code were ignored.

[0091] In another example protocol, the magic state extraction method includes an outer code, which is a second-order triorthogonal CCZ code represented by [[8,3,2]]. Here, n=8, k=3, and d=2. This protocol is based on a generalization of triorthogonal codes (Campbell, supra) (see, for example, Brayvi, supra), which involves introducing a logical CCZ gate on the transverse T gate. The steps of the protocol are as follows:

[0092] 1) In Initialize 8 data qubits.

[0093] 2) Measure the X-stable quantum XXXXXXXX on these 8 data qubits.

[0094] Measuring three logical operators X1 = IXIXIXIX, X2 = IIXXIIXX, X3 = IIIIXXXX and

[0095] In response to a result of -1 for either the X stabilizer or the X logic operator, Pauli correction is applied using the ZIIIIIIII and Z logic operators to make the resulting state a logic state.

[0096] 3) Measure IXXIXIIX and perform a post-selection based on the result of +1.

[0097] 4) Applications in data qubits

[0098] 5) In state |000> o1,o2,o3 Initialize three output qubits.

[0099] 6) Measure X o1 X1 twice, the result is x1,x′1.

[0100] Measure X o2 X² twice, the result is x², x′².

[0101] Measure X o3 X3 twice, the result is x3,x′3.

[0102] For the consistent results x1=x′1, x2=x′2, x3=x′3, perform a subsequent selection.

[0103] If x1 = -1, then apply Z. o1 ,

[0104] If x² = -1, then apply Z. o2 ,and

[0105] If x3 = -1, then apply Z. o3 .

[0106] 7) Measuring the X stabilizer on data qubits And then select from the results of +1.

[0107] 8) Destructively measure all data qubits on the Z basis to obtain the result z1,...,z8, and perform a post-selection on all four conditions z1z2z3z4=+1, z2z4z6z8=+1, z3z4z7z8=+1, and z5z6z7z8=+1.

[0108] Here, steps 1, 2, and 3 complete the preparation of three encoded noisy magic states from the eight data qubits. In step 4, the Clifford stabilizer is measured on the eight data qubits. In step 5, the three output qubits are initialized. In step 6, teleportation is initiated, teleporting the three encoded magic states to the three output qubits. In step 7, the X stabilizer is measured on the eight data qubits, in addition to post-selection based on the results. In step 8, the data qubits are destructively measured using the Z stabilizer, and post-selection conditions are applied to the eight data qubits to complete the teleportation. The three output qubits contain extracted |CCZ> states. In step 4, the product of and is selected to ensure that the underlying triorthogonal encoding satisfies “three-level” orthogonality (see, for example, J. Haah, Phys. Rev. A 97, 04237, 2018, arXiv: 1709.08658, hereinafter referred to as Haah), which eliminates the need for Campbell, supra’s Clifford correction, ibid.

[0109] The depth counting is similar to the previous protocol, counting only multi-qubit measurements. Here, a "double-fault-tolerant" cat state is required. There are 4 + 1 + 1 = 6 measurements in steps 2, 3, and 6 that do not involve output qubits. There are 6 measurements in step 5 that involve output qubits. Using surface patches with distances of d (data) and d′ (output), the duration of this protocol is 6.9d + 6.(7d + 2d′). The number of qubits used is 4.16d. 2 +5d′ 2 The appendages of the surface code were ignored.

[0110] The error model will now be discussed. The complete protocol is simulated using density matrices, which is straightforward since they involve at most 11 qubits. For p out The numerical check is given by p, p as D. t ∈(10 -6 10 -4 The function is executed and fitted to a polynomial formula. or Where a, b, c, and d are the fitting parameters. Table 1 shows the rounded integers of these coefficients.

[0111]

[0112] Table 1. Tracking distance The measured output state error is used as two parameters p, p t The function, p, p t These represent the noise intensity of the Clifford operation and the T-gate / state, respectively.

[0113] The example CCZ state extraction protocol might be designed to implement quadratic error suppression, and therefore all cat states there might be configured to tolerate only quadratic errors (see, for example, reference 14). Assuming the cat states do indeed tolerate only quadratic errors, then analyzing step 7, which should implement a protocol to suppress quadratic errors for X, makes little sense. Therefore, in order to extract p from the CCZ state protocol... out A conservative estimate was made, and the simulation used an alternative to step 7.

[0114] 7a) Destructively measure all data qubits on the Z basis, resulting in z1,...,z8. Perform a post-selection under the condition z1z2z3z4z5z6z7z8=+1. If z7z8=-1, apply X. o1 If z6z8 = -1, apply X. o2 If z4z8 = -1, apply X. o3 .

[0115] In practice, step 7 might be preferable to replacing step 7a because it may capture more error without complex quantum operations compared to step 7a. The overall success probability decreases when we perform step 7 instead of 7a, but the decrease is negligible. If anyone is curious, under our noisy model, using step 7 instead of 7a, p... out How much will it be? The formula below is provided:

[0116] if if And if D(ρ) = XρX, then

[0117] In order to address the p proposed in test table I out The accuracy of the leading-order formula, i.e. the contribution of higher-order terms, is determined by setting p = 10. -2 λ and p t =10 -2 (1-λ), calculate p out As a function of λ = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0, it can be observed that in all cases, for these p... out The value of p out The relative accuracy of the preamble order formula is 29%. Note that in this analysis, the traditional preamble order formula was reproduced by setting p = 0 and D(ρ) = ZρZ.

[0118] (i) For [[6;2;2]] (C. Jones, Phys. Rev. A 87, 022328, 2013, arXiv: 1212.5069, hereinafter referred to as Jones), which is equivalent to the smallest example of (Haah, ibid.) triorthogonal codes (Brayvi, ibid.).

[0119] (ii) For [[7;1;3]] (Knill), which is equivalent to the smallest example of the (Haah, ibid.) quantum Reed-Muller code (S. Brayvi and A. Kitaev, Phys. Rev. A 71, 022316, 2005, arXiv: quant-ph / 0403025; and J. Haah and M. Hastings, Quantum 2, 71, 2018, arXiv: 1709.02832, hereinafter referred to as Haah 2018B), and

[0120] (iii) For [[8;3;2]] (See, for example, Campbell, ibid.; B. Eastin, Phys. Rev. A, 87, 032321, 2013, arXiv:1212.4872; Jones, ibid.; and Haah 2018B, ibid.)

[0121] Acceptance probability p accept This was also derived through numerical calculations, and the formula is satisfied in all three error channels considered. (1-p) and (1-p t The exponent of ) is approximately the number of possible error locations under our error model. Note that this acceptance probability assumes that the cat-state preparation is successful.

[0122] In some examples, the surface codes in the patch are configured to operate in error correction mode, where errors are detected and corrected. In these examples, the outermost code can be configured to operate in error detection mode, where the extracted magic state is retained for quantum computation if no error is detected, and discarded if one or more errors are detected. In other examples, a partial error correction mode can be used, in which case the error can be corrected if a small number of errors give the observed syndrome, otherwise it is discarded. More generally, a set of observed syndromes can be selected for correction, while others are selected for discarding. Since it involves surface codes within a magic state factory, if a state is discarded due to an error, it may have no effect on the rest of the computation. However, if a state on an observed syndrome is discarded, it may need to discard all computation up to that point due to entanglement with other states.

[0123] In other examples, the surface code in the patch of the magic-state factory can be configured to operate in error detection mode. In such an example, a surface code at a distance of d can suppress errors up to order d by performing d rounds of syndrome measurements after each logic measurement. If fewer than d physical errors occur in any round of logic measurements, the logic error can be suppressed. The average or error is p times the number of error locations; there are d... 2 -1 syndrome, therefore d(d) can be performed. 2 -1) Synthetic measurements. Depending on the physical implementation and the error rate of the physical qubits, each synthetic measurement may be decomposed into several physical operations. It may require pd(d 2 -1) << 1 yields a high probability that the state will not be discarded in a given round, the specific value depending on the implementation of the syndrome measurement. As an example, the [[7,1,3]] code may require approximately ~60 patches. Considering a total of ~100 rounds, this method may require 6000d(d 2 -1)≈1.610 5 <<p -1 Achieve significant throughput. In some examples, this is possible if the physical error rate of the qubit is small, for example, less than 10^6 qubits. -6 It may not require a surface code patch.

[0124] In other examples, the surface code can be configured to operate in a partial error detection mode. In these examples, the code can be configured to correct errors with lower weights and discard the state when a higher-weight error is detected. However, if multiple errors are detected, i.e., higher-weight errors are detected, the qubit can be discarded. In some examples, the code can be configured to correct at most one error in each patch of each round. In such examples, the method can achieve... At high physical noise rates, it may be useful to implement a more relaxed form of partial error correction, in which case a larger number of errors can be corrected.

[0125] Figure 6A flowchart depicting an example method 600 for extracting magic states, the magic states being encoded in a [[n,k,d]] block code including an outer code, is shown. Method 600 includes, at 602, preparing K encoded noisy magic states using N data qubits. In some examples, the outer code may include a weakly self-determined block code represented by [[6,2,2]] or [[7,1,3]]. In other examples, the outer code may include a second-order triorthogonal CCZ code represented by [[8,3,2]]. Method 600 further includes, at 604, measuring the Clifford stabilizer on the N data qubits, thereby applying an internal error correction code to the N data qubits. Method 600 further includes, at 606, initializing the K output qubits, and at 608, initiating the teleportation of the K extracted magic states derived from the K encoded noisy states to the K output qubits. Continuing, method 600 includes, at 610, measuring the x-stabilizer on the N data qubits and performing a post-selection on all +1 results. Method 600 further includes, at 612, destructively measuring each data qubit using the Z-stabilizer and applying one or more post-selection conditions to the N data qubits to complete the teleportation of K extracted magic states to K output qubits. After completing the teleportation of the K extracted magic states to the N data qubits, method 600 includes, at 614, using the K extracted magic states in quantum computing.

[0126] In some embodiments, the methods and processes described herein may be associated with a computing system of one or more computing devices. In particular, such methods and processes may be implemented as computer applications or services, application programming interfaces (APIs), libraries, and / or other computer program products.

[0127] Figure 7 A non-limiting embodiment of a computing system 700 is illustrated schematically, which may implement one or more of the methods and processes described above. The computing system 700 is shown in a simplified form. The computing system 700 may take the form of one or more personal computers, server computers, tablet computers, home entertainment computers, network computing devices, gaming devices, mobile computing devices, mobile communication devices (e.g., smartphones), and / or other computing devices.

[0128] The computing system 700 includes a logic subsystem 702 and a storage subsystem 704. The computing system 700 may optionally include a display subsystem 706, an input subsystem 708, a communication subsystem 710, and / or... Figure 7 Other components not shown.

[0129] The logic subsystem 702 includes one or more physical devices configured to execute instructions. For example, the logic subsystem may be configured to execute instructions that are part of one or more applications, services, programs, routines, libraries, objects, components, data structures, or other logical structures. Such instructions may be implemented to perform tasks, implement data types, transform the state of one or more components, achieve technical effects, or otherwise achieve a desired result.

[0130] The logical subsystem 702 may include one or more processors configured to execute software instructions. Alternatively or additionally, the logical subsystem may include one or more hardware or firmware logical subsystems configured to execute hardware or firmware instructions. The processor of the logical subsystem may be single-core or multi-core, and the instructions executed thereon may be configured for sequential, parallel, and / or distributed processing. Individual components of the logical subsystem may optionally be distributed across two or more separate devices that may be remotely located and / or configured for coordinated processing. Aspects of the logical subsystem may be virtualized and executed by remotely accessible, networked computing devices configured in a cloud computing configuration.

[0131] Storage subsystem 704 includes one or more physical devices configured to store instructions executable by a logical subsystem to implement the methods and processes described herein. When such methods and processes are implemented, the state of storage subsystem 704 can be transformed—for example, to store different data.

[0132] Storage subsystem 704 may include removable and / or built-in devices. Storage subsystem 704 may include optical memory (e.g., CD, DVD, HD-DVD, Blu-ray disc, etc.), semiconductor memory (e.g., RAM, EPROM, EEPROM, etc.), and / or magnetic memory (e.g., hard disk drive, floppy disk drive, magnetic tape drive, MRAM, etc.), etc. Storage subsystem 704 may include volatile, non-volatile, dynamic, static, read / write, read-only, random access, sequential access, location-addressable, file-addressable, and / or content-addressable devices.

[0133] It is understood that the storage subsystem 704 includes one or more physical devices. However, aspects of the instructions described herein are alternatively propagated by a communication medium (e.g., electromagnetic signals, optical signals, etc.) that are not stored by the physical device for a limited duration.

[0134] Aspects of the logic subsystem 702 and the storage subsystem 704 can be integrated together into one or more hardware-logic components. For example, such hardware-logic components may include field-programmable gate arrays (FPGAs), application-specific integrated circuits (PASICs / ASICs), application-specific standard products (PSSPs / ASSPs), system-on-a-chip (SOCs), and complex programmable logic devices (CPLDs).

[0135] As is understood, the term "service" as used in this article refers to an application that can execute across multiple user sessions. A service can be used for one or more system components, programs, and / or other services. In some implementations, a service can run on one or more server-computing devices.

[0136] When included, the display subsystem 706 can be used to present a visual representation of the data stored by the storage subsystem 704. This visual representation may take the form of a graphical user interface (GUI). Since the methods and processes described herein change the data stored by the storage subsystem, the state of the display subsystem 706 can also be changed to visually represent the changes in the underlying data, as the state of the storage subsystem is transformed. The display subsystem 706 may include one or more display devices utilizing virtually any type of technology. Such display devices may be combined with the logic subsystem 702 and / or the storage subsystem 704 in a shared enclosure, or such display devices may be peripheral display devices.

[0137] When included, the input subsystem 708 may include or interface with one or more user input devices, such as a keyboard, mouse, touchscreen, or game controller. In some embodiments, the input subsystem may include or interface with selected Natural User Input (NUI) components. Such components may be integrated or peripheral, and the transduction and / or processing of input actions may be performed on-board or off-board. Example NUI components may include a microphone for speech and / or voice recognition; an infrared, color, stereo, and / or depth camera for machine vision and / or gesture recognition; a head tracker, eye tracker, accelerometer, and / or gyroscope for motion detection and / or intent recognition; and an electric field sensing component for assessing brain activity.

[0138] When included, the communication subsystem 710 can be configured to communicatively couple the computing system 700 to one or more other computing devices. The communication subsystem 710 may include wired and / or wireless communication devices compatible with one or more different communication protocols. As a non-limiting example, the communication subsystem may be configured to communicate via a wireless telephone network, a wired local area network, a wireless local area network, or a wide area network. In some embodiments, the communication subsystem may allow the computing system 700 to send and / or receive messages from other devices via a network such as the Internet.

[0139] Another example provides a method for extracting magic states on a quantum computing device, where the magic states are encoded in a block code of [[n,k,d]] including an outer code. This method involves preparing k encoded noisy magic states using n data qubits; measuring the Clifford stabilizer on the n data qubits to apply an internal error correction code to the n data qubits; initializing the k output qubits; initiating the teleportation of the k extracted magic states derived from the k encoded noisy magic states to the k output qubits; measuring the X stabilizer on the n data qubits and performing a post-selection on all +1 results; destructively measuring each data qubit using the Z stabilizer and applying one or more post-selection conditions to the n data qubits to complete the teleportation of the k extracted magic states to the k output qubits; and using the k extracted magic states in quantum computing. In some such examples, the outer code can be a second-order normal weak self-dual CSS code represented by [[6,2,2]], where n=6, k=2, and d=2. In such an example, k encoded noisy magic states are prepared using n qubits, including... Initialize 6 data qubits and... and Placed on one side of the qubit array, XXXXII and IIXXXX are measured on 6 data qubits, and Z measurement is applied to the 6 data qubits to obtain logic qubits including logic states with ZIZIZI≈IZIZIZ≈+1, and via measurement X... t1 X1, X t2 X2, then measure Z t1 Z t2 Then, Pauli Z-correction is performed on the logical qubits to teleport the states in qubits t1 and t2 to the logical qubits. Pauli Z-correction is based on X... t1 X1, X t2 X2 and Z t1 Z t2 The measurement of X1 and X2 is performed by logical operators X1 = XIXIXI and X1 = XIXIXI; the measurement of the Clifford stabilizer on n data qubits includes the application of data qubits. Measurement on data qubits The process involves two rounds of selection and post-selection of all +1 results, as well as applications on data qubits. Initializing k output qubits includes Initialize two output qubits o1 and o2; initiate the teleportation of the k encoded magic states to the k output qubits, including measuring X. o1 X1 twice, the result is x1,x′1, measure X o2X2 twice, resulting in x2, x′2. For the consistent results x1=x′1 and x2=x′2, the next choice is made. If x1=-1, then Z is applied. o1 If x2 = -1, then apply Z. o2 ;Measure the X stabilizer on n data qubits and perform post-selection on all +1 results, including measuring XXXXII and IIXXXX on 6 data qubits and performing post-selection on all +1 results; Destructively measure each data qubit using the Z stabilizer and apply one or more post-selection conditions to the n data qubits to complete the teleportation from k extracted magic states to k output qubits, including destructively measuring individual data qubits on the Z basis to obtain results z1,...z6, and performing post-selection on both the conditions z1z2z3z4=+1 and z3z4z5z6=+1. If z1z3z5=-1, then apply X. o1 If z2z4z6 = -1, then apply X. o2 ; and k extracted magic states containing 2 T states. In some such examples, the outer code can be a third-order normal weak self-dual CSS code represented by [[7,1,3]], where n=7, k=1 and d=3. In such examples, the k encoded noisy magic states are prepared using n qubits, including Initialize 7 data qubits and... Placed on one side of the qubit array, three X-stabilizers XIXIXIX, IXXIIXX, and IIIXXXX are measured on 7 data qubits, and a Z-measurement is applied on 6 data qubits to obtain logical qubits consisting of logical states including ZZZZZZZ≈+1, and via the measurement of X t X and then Z are measured. t Invisibly transmitted into the code, the measurement X is a logical operator. The method further includes measuring the three X stabilizers and performing post-selection on all +1 results; measuring the Clifford stabilizer on n data qubits includes applying... Measurement on data qubits The process involves two rounds of selection and post-selection of all +1 results, as well as applications on data qubits. The k output qubits are initialized in the state |0>o; the teleportation from the k encoded magic states to the k output qubits involves measuring the three equivalent X logic operators X of the code multiplication X operator on the output qubits. o (IXIXIXI), X o (XIIXXII), X o(XXXIIII), the result is x1,x2,x3=±1. The next selection is made based on the consistent result x=x1=x2=x3. If x=-1, then Z is applied. o Measuring the X stabilizer on n data qubits and performing post-selection on all +1 results includes measuring three X stabilizers and performing post-selection on all +1 results; and destructively measuring each data qubit using the Z stabilizer and applying one or more post-selection conditions to the n data qubits to complete the teleportation from the K extracted magic states to the K output qubits, including destructively measuring all data qubits on the Z basis to obtain results z1,...z7, and performing post-selection on three conditions z1z3z5z7=+1, z2z3z6z7=+1, and z4z5z6z7=+1. If z1z3z5=-1, then X is applied. o Furthermore, the k extracted magic states include one T state. In some such examples, the outer code may include a second-order triorthogonal CCZ code denoted by [[8,3,2]], where n=8, k=3, and d=2. In such examples, the k encoded noisy magic states are prepared using n qubits, including... Initialize 8 data qubits. Measure the X-stableton XXXXXXXX on the 8 data qubits, measure the three X-logic operators X1 = IXIXIXIX, X2 = IIXXIIXX, X3 = IIIIXXXXX, and in response to the occurrence of a -1 result for any term of the X-stableton and X-logic operators, apply Pauli correction through the ZIIIIIIII and Z-logic operators to make the resulting state a logical state. And measuring IXXIXIIX and performing post-selection on the +1 result; measuring the Clifford stabilizer on n data qubits, including applications on data qubits. Initializing k output qubits includes state |000> o1,o2,o3 Initialize 3 output qubits; initiate k encoded magic states for teleportation to the k output qubits, including measuring X. o1 X1 twice, the result is x1,x′1, measure X o2 X2 twice, the result is x2,x′2, measure X o3 X3 twice, resulting in x3, x′3. For the consistent results x1=x′1, x2=x′2, x3=x′3, perform a second selection. If x1=-1, then apply Z. o1 If x² = -1, then apply Z. o2 If x3 = -1, then apply Z. o3 ;Measure the X stabilizer on n data qubits and perform post-selection on all +1 results, including measuring the X stabilizer on the data qubits. The +1 result is then subjected to post-selection; each data qubit is destructively measured using the Z-stabilizer, and one or more post-selection conditions are applied to the n data qubits to complete the teleportation from the k extracted magic states to the k output qubits. This includes destructively measuring all data qubits on the Z-basis to obtain the result z1,...,z8, and performing post-selection on all four conditions z1z2z3z4=+1, z2z4z6z8=+1, z3z4z7z8=+1, and z5z6z7z8=+1; and the k extracted magic states include 3 qubits in the |CCZ> state. In some such examples, applying the Z measurement and applying the X measurement may each include performing a Pauli framework update. In some such examples, the [[n,k,d]] block code may additionally or alternatively include 8n qubits forming a 4×(2n) rectangle, where the first row is occupied by S-states; the second row by T-states; the third row by data qubits of every other column of the block code; and the fourth row reserved for cat state preparation. In some such examples, the method may additionally or alternatively include the preparation of the cat states, including the preparation of |00> ab X was measured using Pauli correction. a X b , to obtain |00> ab +|11> ab Shrink b to match the patch size in the fourth row of a 4×(2n-1) rectangle, and fully measure the ZZ between the nearest neighbors in the fourth row, discarding every other qubit. Preparing k encoded noisy magic states using n data qubits may involve multi-qubit Pauli operations; initiating teleportation from the k encoded magic states to the k output qubits may include multi-qubit Pauli operations; and multi-qubit Pauli operations may be performed via cat states. In some such examples, the inner code may additionally or alternatively be configured to operate in error detection mode, and the method may include post-selection based on Clifford stabilizer measurements. In some such examples, the inner code may additionally or alternatively be configured to operate in partial error correction mode, and the method may include determining the weights of errors associated with Clifford stabilizer measurements, correcting errors with smaller weights, and performing post-selection based on higher weights, thereby not using the k extracted magic states in quantum computing. In some such examples, preparing k encoded noisy magic states using n data qubits may include injecting a T state via a horizontal ZZ measurement and a single-qubit X measurement, thereby teleporting the T state to a qubit horizontally adjacent to the data qubits. In some such examples, the output may additionally or alternatively include k+2 output qubits, comprising k high-quality magic states and 2 appendages.

[0140] Another example provides a quantum computer comprising a qubit register containing multiple qubits; a modulator configured to perform one or more quantum logic operations on the multiple qubits; a demodulator configured to output data exposing the quantum states of the multiple qubits; a controller operatively coupled to the modulator and demodulator; and a computer memory storing instructions that cause the controller, within a [[n,k,d]] block code, to prepare k encoded noisy magic states using n data qubits of the multiple qubits; measure the Clifford stabilizer on the n data qubits; initialize k output qubits of the multiple qubits; initiate teleportation from the k output qubits derived from the k encoded magic states to the k encoded magic states; measure the X stabilizer on the n data qubits and perform post-selection on all +1 results; destructively measure each data qubit using the Z stabilizer and apply one or more post-selection conditions to the n data qubits to complete the teleportation from the k extracted magic states to the k output qubits; and use the k extracted magic states in quantum computing. In some such examples, the k extracted magic states may include one or more T states. In some such examples, the k extracted magic states may additionally or alternatively include three qubits in a CCZ state. In some such examples, the [[n,k,d]] block code may additionally or alternatively include 8n qubits forming a 4×2n rectangle, comprising a first row occupied by S states, a second row occupied by T states, a third row occupied by data qubits every other column, and a fourth row reserved for cat state preparation. In some such examples, the quantum computer may additionally or alternatively be a topological quantum computer. In some such examples, the quantum computer may additionally or alternatively include Majorana lines, and the qubits may be Majorana fermions within the Majorana lines.

[0141] Another example provides a method for extracting magic states on a quantum computer, the magic states being encoded in a block code of [[n,k,d]] including an outer code. This method includes initializing n data qubits in a data qubit array; placing k noisy magic states on one side of the data qubit array; performing a stabilizer measurement on the k data qubits; performing one or more logic operator measurements on the k noisy magic states and the n data qubits, thereby teleporting the state of the k noisy magic states to the k encoded magic states; performing one or more Clifford stabilizer measurements on the n data qubits, thereby applying an internal error correction code to the n data qubits; initializing k output qubits; initiating the teleportation of the k encoded magic states to the k output qubits; performing an X stabilizer measurement on the n data qubits; performing a destructive Z stabilizer measurement on the data qubits to complete the teleportation of the k encoded magic states to the k output qubits, which now contain the k extracted magic states; and using the k extracted magic states in a quantum computing operation. In some such examples, k noisy magic states can be k noisy T states; k encoded magic states can be k encoded T states; and k extracted magic states can be k high-fidelity T states. In some such examples, k noisy magic states can be k noisy CCZ states, k encoded magic states are k encoded T states, and k extracted magic states can be k high-fidelity CCZ states.

[0142] It should be understood that the configurations and / or methods described herein are exemplary in nature, and these specific embodiments or examples should not be considered limitingly, as many variations are possible. The particular routines or methods described herein may represent one or more of any number of processing strategies. Therefore, the various behaviors illustrated and / or described may be performed in the illustrated and / or described order, other orders, in parallel, or omitted. Similarly, the order of the above processing may be changed.

[0143] The subject matter of this disclosure includes all novel and non-obvious combinations and sub-combinations of the various processes, systems and configurations disclosed herein, as well as any and all equivalents.

Claims

1. A method for extracting magic states on a quantum computing device, the magic states being encoded in a [[n,k,d]] block code including an outer code, where d is the code distance, the method comprising: Use n data qubits to prepare k encoded noisy magic states; The Clifford stabilizer is measured on the n data qubits, thereby applying an internal error correction code to the n data qubits; Initialize k output qubits; Initiate the teleportation of k extracted magic states derived from the k encoded noisy magic states to the k output qubits; The X stabilizer is measured on the n data qubits, and a post-selection is performed on all +1 results; Each data qubit is destructively measured using the Z-stableton, and one or more post-selection conditions are applied to the n data qubits to complete the teleportation of the k extracted magic states to the k output qubits; as well as The k extracted magic states are used in quantum computing.

2. The method according to claim 1, wherein the outer code is a second-order normal weak self-dual CSS code represented by [[6,2,2]], where n=6, k=2, and d=2; The preparation of k encoded noisy magic states using n qubits includes: 1) In Initialize 6 data qubits and... and Placed on one side of the qubit array, 2) Measurement on the 6 data qubits XXXXII and IIXXXX And apply Z measurement to the 6 data qubits to obtain including those with ZIZIZI ≈ IZIZIZ Logical qubits with a logical state of ≈+1, and 3) Through measurement , Then measure , Then, Pauli Z-correction is performed on the logical qubits to teleport the states in qubits t1 and t2 to the logical qubits. The Pauli Z-correction is based on... , and , Measurement, and It is a logical operator and ; in Measuring the Clifford stabilizer on the n data qubits includes: 4) Application on the data qubits , 5) Measurement on the data qubits Twice, and then select the last result for all +1 results, and 6) Applying to the data qubits ; The initialization of k output qubits includes: 7) In Initialize two output qubits ; The teleportation from the k encoded magic states to the k output qubits includes: 8) Measurement Twice, the result was ; Measurement Twice, the result was ; For consistent results and To conduct a subsequent selection, and if ,application , and if ,application ; The measurement of the X stabilizer on the n data qubits and the subsequent selection of all +1 results include: 9) Measurement on the 6 data qubits XXXXII and IIXXXX And select the last result from all results that are +1; The method of destructively measuring each data qubit using the Z-stableton and applying one or more post-selection conditions to the n data qubits to complete the teleportation from the k extracted magic states to the k output qubits includes: 10) Destructively measure individual data qubits on the Z-basis to obtain results. and under the conditions and Choose the latter from both, and if ,application , and if ,application ;as well as The k extract magic states include 2 T states.

3. The method according to claim 1, wherein the outer code is a third-order normal weak self-dual CSS code represented by [[7,1,3]], where n=7, k=1, and d=3; The preparation of k encoded noisy magic states using n qubits includes: 1) In Initialize 7 data qubits and... Placed on one side of the qubit array, 2) Measure three X stabilizers on the seven data qubits. XIXIXIX , IXXIIXX and IIIXXXX And apply the Z measurement to the 6 data qubits to obtain including having The logical qubits of the logical state, and 3) Through measurement And then measure Invisibly transmitted to the code, measured It is a logical operator ; Further includes: 4) Measure the three X stabilizers and perform post-selection on all +1 results; The measurement of the Clift stabilizer on the n data qubits includes: 5) Application on the data qubits , 6) Measurement on the data qubits Twice, and then select the last result for all +1 results, and 7) Application on the data qubits ; The initialization of k output qubits includes: 8) In the stated state Initialize the output qubits; The teleportation from the k encoded magic states to the k output qubits includes: 9) Measure the three equivalent X logic operators of the code multiplied by the X operator on the output qubit. , , The result is , Consistent results The above is the next selection, and if Then apply ; The measurement of the X stabilizer on the n data qubits and the subsequent selection of all +1 results include: 10) Measure the three X stabilizers and perform post-selection on all +1 results; and The teleportation from the K extracted magic states to the K output qubits by destructively measuring each data qubit using the Z-stableton and applying one or more post-selection conditions to the n data qubits includes: 11) Destructively measure all data qubits in the Z-basis to obtain the results. And under 3 conditions , and Select the next option. if ,application ,as well as The k extract magic states include one T state.

4. The method according to claim 1, wherein the outer code is a second-order triorthogonal CCZ code represented by [[8,3,2]], where n=8, k=3, and d=2; The preparation of k encoded noisy magic states using n qubits includes: 1) In Initialize 8 data qubits. 2) Measure the X stabilizer on the 8 data qubits. XXXXXXXX , Measuring 3 X logic operators , , ,as well as In response to the occurrence of a -1 result for either the X stabilizer or the X logic operator, by ZIIIIIIII Applying Pauli correction to the Z-logic operator results in the state being the stated logic state. ,as well as 3) Measurement IXXIXIIX And then perform a post-selection on the result of +1; The measurement of the Clifford stabilizer on the n data qubits includes: 4) Application on the data qubits ; The initialization of k output qubits includes: 5) In the stated state Initialize 3 output qubits; The teleportation from the k encoded magic states to the k output qubits includes: 6) Measurement Twice, the result was , Measurement Twice, the result was , Measurement Twice, the result was , For consistent results , , To proceed with the selection process, if Then apply , if Then apply ,as well as if Then apply ; The measurement of the X stabilizer on the n data qubits and the subsequent selection of all +1 results include: 7) Measure the X stabilizer on the data qubit. And select the next result after adding 1; The teleportation from the k extracted magic states to the k output qubits involves destructively measuring each data qubit using the Z-stableton and applying one or more post-selection conditions to the n data qubits. 8) Destructively measure all data qubits in the Z-basis to obtain the results. and in all four conditions , , and The above will be used for subsequent selection; and The k extracted magic states include those in |CCZ The state consists of 3 qubits.

5. The method of claim 1, wherein applying the Z-measure and applying the X-measure each comprise performing a Pauli framework update.

6. The method according to claim 1, wherein the [[n, k, d]] block code comprises 8n qubits, the 8n qubits forming a 4×(2n) rectangle, wherein the first row is occupied by S-states; the second row is occupied by T-states; the third row is occupied by data qubits of every other column of the block code; and the fourth row is reserved for cat state preparation.

7. The method according to claim 6, further comprising cat-state preparation, wherein the cat-state preparation comprises: 1) Preparation And measured using Pauli correction. In order to obtain ; 2) Shrink b to match the size of the patch in the fourth row of the 4×(2n-1) rectangle, and 3) Fully measure the ZZ of the nearest neighbors in the fourth row and discard every other qubit; The preparation of k encoded noisy magic states using n data qubits includes multi-qubit Pauli operations; The teleportation from the k encoded magic states to the k output qubits includes multi-qubit Pauli operations; and The multi-qubit Pauli operation is performed via the cat state.

8. The method of claim 1, wherein the internal code is configured to operate in error detection mode, and wherein the method further comprises performing a post-selection based on the Clifford stabilizer measurement.

9. The method of claim 1, wherein the internal code is configured to operate in a partial error correction mode, and wherein the method further comprises: Determine the weights of the errors associated with the Clifford stabilizer measurement; For errors with smaller weights, correct the error; as well as For errors with higher weights, a post-selection is performed based on the error, so that the k extracted magic states are not used in quantum computing.

10. The method of claim 1, wherein preparing k encoded noisy magic states using n data qubits comprises: The T-state is teleported to a qubit horizontally adjacent to the data qubit by injecting a horizontal ZZ measurement and a single-qubit X measurement.

11. The method of claim 1, wherein the output comprises k+2 output qubits, the k+2 output qubits comprising k high-quality magic states and 2 appendages.

12. A quantum computer, comprising: A qubit register, comprising multiple qubits; The modulator is configured to perform one or more quantum logic operations on the plurality of qubits; A demodulator is configured to output data that exposes the quantum states of the plurality of qubits; A controller is operatively coupled to the modulator and the demodulator; as well as Computer memory stores instructions that cause the controller to: Within the [[n,k,d]] block code, k encoded noise magic states are prepared using n data qubits of the plurality of qubits, where d is the code distance; The Clifford stabilizer is measured on the n data qubits; Initialize the k output qubits of the plurality of qubits; Initiate the teleportation of the k encoded magical states derived from the k encoded magical states to the k output qubits; The X stabilizer is measured on the n data qubits, and all +1 results are then selected. Each data qubit is destructively measured using the Z-stableton, and one or more post-selection conditions are applied to the n data qubits to complete the teleportation from the k extracted magic states to the k output qubits; as well as The k extracted magic states are used in quantum computing.

13. The quantum computer of claim 12, wherein the k extracted magic states comprise one or more T states.

14. The quantum computer of claim 13, wherein the quantum computer is a topological quantum computer.

15. The quantum computer of claim 12, wherein the k extracted magic states comprise 3 qubits in the CCZ state.

16. The quantum computer of claim 12, wherein the [[n, k, d]] block code comprises 8n qubits of the plurality of qubits, the 8n qubits forming a 4×2n rectangle, the 4×2n rectangle comprising: The first row is occupied by S-states. The second row is occupied by the T state. The third row is occupied by data qubits that alternate every other column, and The fourth line is reserved for the preparation of the cat state.

17. The quantum computer of claim 12, further comprising a Majorana line, wherein the plurality of qubits are Majorana fermions within the Majorana line.

18. A method for extracting a magic state on a quantum computer, the magic state being encoded in a block code of [[n,k,d]] including an outer code, where d is the code distance, the method comprising: Initialize n data qubits in the data qubit array; Place k noisy magic states on one side of the data qubit array; Perform a stable sub-measurement on k data qubits; One or more logic operator measurements are performed on the k noisy magic states and n data qubits to teleport the state of the k noisy magic states to the k encoded magic states; Perform one or more Clifford stabilizer measurements on the n data qubits to apply an internal error correction code to the n data qubits; Initialize k output qubits; Initiate the teleportation of the k encoded magic states to the k output qubits; Perform X-stabilizer measurements on the n data qubits; A destructive Z-stabilizer measurement is performed on the data qubits to complete the teleportation of the k encoded magic states to the k output qubits, which now comprise k extracted magic states; and The k extracted magic states are used in quantum computing.

19. The method of claim 18, wherein the k noise magic states are k noise T states, wherein the k encoded magic states are k encoded T states, and wherein the k extraction magic states are k high-fidelity T states.

20. The method of claim 18, wherein the k noise magic states are k noise CCZ states, wherein the k encoding magic states are k encoding T states, and wherein the k extraction magic states are k high-fidelity CCZ states.

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