Measuring the purity of quantum states

Abstract

Methods, systems, and apparatus for measuring the purity of quantum states. In one aspect, a method for determining the average purity of a plurality of output quantum states, wherein the plurality of output quantum states correspond to the application of corresponding random quantum circuits of the same circuit depth to the same initial quantum state, the method comprising: obtaining a plurality of data terms, wherein each data term corresponds to a corresponding random quantum circuit of the same circuit depth and represents the probability of applying the corresponding random quantum circuit to the initial quantum state to produce a corresponding measurement result; calculating the variance of the plurality of data terms; determining a Porter-Thomas distribution having a dimension equal to the dimension of each output quantum state; and dividing the calculated variance by the variance of the Porter-Thomas distribution to determine the average purity.

Description

Technical Field

[0001] This explanation relates to quantum computing. Background Technology

[0002] Quantum computing uses quantum mechanical phenomena such as superposition and entanglement to perform computations. A quantum circuit is an example model for quantum computing, where computation is a series of quantum logic gates that are reversible transformations on a quantum mechanical simulation of an n-bit register. Summary of the Invention

[0003] This manual describes techniques for measuring the purity of quantum states.

[0004] Generally, an innovative aspect of the subject matter described in this specification can be implemented in a method for determining the average purity of a plurality of output quantum states, wherein the plurality of output quantum states correspond to the application of corresponding random quantum circuits of the same circuit depth to the same initial quantum state, the method comprising: obtaining a plurality of data items, wherein each data item corresponds to a corresponding random quantum circuit of the same circuit depth and represents the probability of applying the corresponding random quantum circuit to the initial quantum state to produce a corresponding measurement result; calculating the variance of the plurality of data items; determining a Porter-Thomas distribution having a dimension equal to that of each output quantum state; and dividing the calculated variance by the variance of the Porter-Thomas distribution to determine the average purity.

[0005] Other embodiments of this aspect include corresponding computer systems, apparatuses, and computer programs recorded on one or more computer storage devices, each configured to perform actions of the method. A system of one or more computers may be configured to perform specific operations or actions by having software, firmware, hardware, or combinations thereof installed on the system, which, in operation, cause the system to perform said actions. A system of one or more computer programs may be configured to perform specific operations or actions by including instructions that, when executed by a data processing device, cause the device to perform said actions.

[0006] The foregoing and other embodiments may optionally include one or more of the following features individually or in combination. In some embodiments, each output quantum state is represented by a depolarization channel having a depolarization channel parameter p, which represents the probability that the output quantum state is a pure state output.

[0007] In some implementations, the depolarization channel parameter p is equal to 1, and the plurality of data items are distributed according to a Porter-Thomas distribution.

[0008] In some implementations, the variance of the Porter-Thomas distribution is equal to... Where D represents the dimension of the Porter-Thomas distribution.

[0009] In some implementations, the depolarization channel parameter p is equal to 0, and the plurality of data items are distributed according to a δ function at 1 / D, where D represents the dimension of the Porter-Thomas distribution.

[0010] In some implementations, the random quantum circuit includes a random quantum circuit generated for cross-entropy benchmark experiments.

[0011] In some implementations, a random quantum circuit includes a quantum circuit comprising one or more quantum gates randomly sampled from a predetermined set of quantum gates.

[0012] In some implementations, each random quantum circuit includes the same number of quantum gates.

[0013] In some implementations, purity includes single-qubit purity, and each random quantum circuit includes multiple single-qubit quantum gates having an error rate within the same predetermined range.

[0014] In some implementations, purity includes n-qubit purity, and each random quantum circuit includes i) a plurality of single-qubit quantum gates having an error rate within the same predetermined range, and ii) the same n-qubit quantum gate.

[0015] In some embodiments, the method further includes: obtaining measurement data corresponding to measurement results of applying a corresponding random quantum circuit of the same circuit depth to the same initial quantum state; and using the obtained measurement data to calculate the probability that applying the corresponding random quantum circuit to the initial quantum state will produce the corresponding measurement result.

[0016] In some embodiments, the method further includes: processing the acquired plurality of data items to determine the quantum state fidelity of the plurality of output quantum states; and calculating the difference between the determined quantum state fidelity and the determined average purity loss, wherein the calculated difference represents a system control error.

[0017] In some implementations, the method further includes: determining one or more adjustments to quantum hardware control parameters based on a calculated difference representing a system control error; and implementing the determined one or more adjustments to perform quantum computing using the quantum computing hardware.

[0018] In some implementations, the method further includes: determining one or more adjustments to quantum hardware control parameters based on a determined average purity; and implementing the determined one or more adjustments to perform quantum computing using the quantum computing hardware.

[0019] The subject matter described in this specification can be implemented in a particular manner to achieve one or more of the following advantages.

[0020] Systems implementing the techniques described herein can determine the purity of a quantum state using a fixed number of experiments. For example, compared to known techniques such as full-state tomography, systems implementing the techniques described herein can determine the purity of a quantum state from raw quantum hardware benchmark data with an exponentially smaller number of pulse sequences. This advantageous scaling allows the techniques described herein for measuring the purity of quantum states to be extended to quantum hardware comprising a large number of qubits.

[0021] Furthermore, the techniques described herein can be implemented as part of a benchmarking process for characterizing the performance of quantum hardware without requiring additional experiments. For example, the techniques described herein can be applied in conjunction with a cross-entropy benchmarking process. Implementing the methods described herein for measuring the purity of quantum states also enables the system to perform error budgeting of the total error to control for errors and decoherence errors.

[0022] Furthermore, the techniques described above can be performed without knowing the specific gate sequence being executed—Porter-Thomas statistics apply as long as benchmark experiments introduce sufficient Hilbert space randomization.

[0023] The techniques described herein can be applied to improve quantum computing hardware and quantum control—key features of high-fidelity quantum computing. For example, tuning that improves the accuracy of existing quantum computing hardware (e.g., the accuracy of quantum computing hardware performing quantum operations) can be determined based on the deterministic purity and error of the quantum states generated by the quantum computing hardware. Furthermore, purity can be used to optimize the parameters of the control models used to implement quantum circuits / quantum gates.

[0024] Details of one or more embodiments of the subject matter of this specification are set forth in the accompanying drawings and the following description. Other features, aspects, and advantages of the subject matter will become apparent from the description, drawings, and claims. Attached Figure Description

[0025] Figure 1 An example system for benchmarking the performance of quantum computing hardware is depicted.

[0026] Figure 2 This is a flowchart of an example process for determining the purity of average quantum states.

[0027] Figure 3 Three graphs are shown, illustrating example measurement data and processed measurement data. Detailed Implementation

[0028] Overview

[0029] Quantum circuits are a model for quantum computing, in which quantum logic gates are applied in a specific sequence to a qubit register to encode quantum information. Theoretically, any quantum algorithm can be implemented with high precision by applying the correct sequence of quantum logic gates. However, in practice, quantum logic gates are prone to error—experiments attempting to implement unitary quantum operations, representing ideal quantum logic gates, often result in noisy quantum operations.

[0030] Due to physical error mechanisms, quantum circuits always contribute some error. It is important to be able to distinguish between systematic errors and errors attributable to noise, as systematic errors can be reduced through careful calibration of the system control. Systematic errors and noise errors can be distinguished by measuring the purity of the output states of the quantum circuit.

[0031] State tomography is commonly used to measure state purity, where the full density matrix is ​​reconstructed and used to quantize state purity. State tomography involves expanding a single cross-entropy benchmark sequence into a set of sequences, each appended with a single-qubit gate. Unfortunately, the full tomographic reconstruction grows exponentially with the number of qubits, depending on the number of sequences required and the number of measurements required per sequence. For example, a state tomography of a single qubit typically requires at least three experiments to constrain the X, Y, and Z directions in a Bloch sphere. A complete n-qubit state tomography typically requires three... n There are 10 experiments. Furthermore, each experiment has N=2... n One output state. If we assume The statistic requires N for each experiment. 2 =2 2n This measurement. Due to double exponential scaling, state tomography becomes difficult to process after a small number of qubits (e.g., for 8 qubits or more).

[0032] This specification describes a technique for measuring quantum state purity using raw measurement data obtained from quantum hardware benchmark experiments. The probability distribution statistics of the measurement data are calculated and used to determine the quantum state purity. Therefore, a fixed total number of experiments, rather than an exponential number, can be used to accurately measure state purity. The determined state purity can be further used to identify system control errors occurring during the corresponding quantum hardware benchmark experiments. Thus, a single quantum hardware benchmark can be used to identify the total error as control error and / or decoherence error.

[0033] In this specification, the term "quantum state purity" is understood to describe the rescaled purity of a quantum state, which is defined as follows:

[0034]

[0035] The rescaling ensures that fully decoherent states have a purity of 0, and pure states have a purity of 1. In equation (1), D represents the Hilbert space size of the quantum system, and ρ represents the quantum state. The purity given by equation (1) can be understood as the square length of the generalized Bloch vector in D dimensions. For example, for a qubit (D = 2), equation (1) gives... <x> 2 + <y> 2 + <z> 2 .

[0036] Example hardware

[0037] Figure 1 An example system for benchmarking the performance of quantum computing hardware is illustrated. Example system 100 is an example of a system implemented as a classical and quantum computer program on one or more classical and quantum computers at one or more locations, wherein the systems, components and techniques described below can be implemented.

[0038] System 100 includes a classical processor 102 that communicates data with quantum computing hardware 104. For convenience, the classical processor 102 and the quantum computing hardware 104 are shown as separate entities; however, in some embodiments, the classical processor 102 may be included within the quantum computing hardware 104. For example, the quantum computing hardware 104 may include one or more components for performing classical computing operations.

[0039] Quantum computing hardware 104 includes components that use quantum circuits to perform quantum computing. For example, quantum computing hardware 104 includes a quantum system 120 and a control device 122. Quantum system 120 includes one or more multilevel quantum subsystems, such as qubits for performing algorithmic operations or quantum computing. The specific implementation of the multilevel quantum subsystems included in quantum computing hardware 104 and how they interact with each other depends on a variety of factors, including the type of quantum computing the quantum computing hardware is performing. For example, multilevel quantum subsystems may include qubits implemented via atomic, molecular, or solid-state quantum systems. In other examples, qubits may include, but are not limited to, superconducting qubits or semiconductor qubits.

[0040] Multilevel quantum systems can be frequency-tunable. For example, each qubit can have an associated operating frequency, which can be adjusted, for example, by applying voltage pulses via one or more drive lines coupled to the qubit using one or more control devices 122. Example operating frequencies include qubit idle frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to the corresponding idle frequency can place the qubit in a state where it does not strongly interact with other qubits and in a state where it can be used to perform a single-qubit gate. As another example, in the case where qubits interact via couplers with fixed coupling, the qubits can be configured to interact with each other by setting their respective operating frequencies to a gate-related frequency that is detuned from their common interaction frequency. In other cases, for example, when qubits interact via tunable couplers, the qubits can be configured to interact with each other by setting the parameters of their respective couplers to achieve interaction between the qubits, and then by setting the respective operating frequencies of the qubits to a gate-related frequency that is detuned from their common interaction frequency. Such interactions can be performed to execute multi-qubit gates.

[0041] The control device 122 may also include a measurement device, such as a readout resonator. The measurement results (measurement data) obtained via the measurement device may be provided to a classical processor included in the quantum computing hardware 104 or to the classical processor 102 for processing and analysis.

[0042] The classical processor 102 receives input data 106 representing the quantum hardware to be benchmarked. For example, the input data 106 may include data representing that the quantum computing hardware 104 is configured to implement quantum logic gates or quantum circuits.

[0043] The classical processor 102 processes the received input data 106 to generate output data 108 representing benchmark results (e.g., characteristics of an implementation of a quantum logic gate or quantum circuit). For example, the output data 108 may include data representing the estimated fidelity and / or purity of the quantum states output during the implementation of the quantum logic gate or quantum circuit by the quantum hardware 104.

[0044] The classical processor 102 includes multiple components for processing received input data. For example, the classical processor 102 may include a random quantum circuit generator 110 and a data processing module 114.

[0045] The random quantum circuit generator 110 can be configured to define random quantum circuits based on the quantum computing hardware 104 and the received input data 106.

[0046] A random quantum circuit is a quantum circuit that includes one or more quantum gates, which are randomly sampled from a predetermined set of quantum gates. The type of random quantum circuit defined by the random quantum circuit generator 110 depends on the benchmark experiment being performed by the system 100.

[0047] For example, when the performance of single-qubit / single-qubit operations is to be benchmarked, the random quantum circuit generator 110 can define multiple random quantum circuits, each comprising one or more corresponding randomly sampled single-qubit gates. For instance, the random quantum circuit generator 110 can be configured to generate a set of predefined single-qubit gates (e.g., including...). A group of T quantum gates, among which This represents a rotation of π / 2 around the X-axis. Let T denote a π / 2 rotation about the y-axis, and let T denote a non-Clifford diagonal matrix {0, e}. iπ / 4 Randomly sample single-qubit gates in the random quantum circuit defined by the random quantum circuit generator 110. In a single-qubit benchmark experiment, the single-qubit gates included in the random quantum circuit can have approximately equal error rates. For example, the error rate of each single-qubit gate in a set of single-qubit gates sampled by the random quantum circuit generator 110 may be from a predetermined range of error rates.

[0048] As another example, when the performance of multi-qubit / multi-qubit operations is to be benchmarked, the random quantum circuit generator 110 can define multiple random quantum circuits, each comprising one or more corresponding randomly sampled single-qubit gates and the same multi-qubit gates. Again, in multi-qubit benchmark experiments, the single-qubit gates included in the random quantum circuits defined by the random quantum circuit generator 110 can have approximately equal error rates.

[0049] Random quantum circuits defined by random quantum circuit generator 110 can have different depths. Random quantum circuit generator 110 can define circuits of different depths by applying multiple clock cycles of gates. That is, random quantum circuit generator 110 can define a random quantum circuit of depth d as equal to d cycles of the same gate sequence. In some embodiments, random quantum circuit generator 110 can define a gate sequence (e.g., including multiple randomly sampled single-qubit gates, then multi-qubit gates) and use the defined gate sequence to define multiple random quantum circuits, where each defined random quantum circuit corresponds to a corresponding number of cycles in the defined gate sequence. For example, in the following... Figure 3 In Figure (a), the random quantum circuit generator 110 defines 30 different gate sequences and defines 500 random quantum circuits corresponding to 1-500 periods of the gate sequence for each gate sequence.

[0050] Quantum circuit 130 is an example of a random quantum circuit generated by random quantum circuit generator 110. Example quantum circuit 130 illustrates a benchmark quantum circuit configured to operate on two qubits q1, q2. Example quantum circuit 130 comprises four cycles, each cycle including two randomly sampled single-qubit gates R1, R2 operating on qubits q1, q2 respectively, and a copy of a two-qubit quantum gate, for example, a CZ gate in this example.

[0051] Classical processor 102 is configured to transmit data 116 representing a defined random quantum circuit to quantum computing hardware 104. Quantum computing hardware 104 is configured to implement the defined random quantum circuit using quantum system 120 and control device 122.

[0052] Quantum computing hardware 104 can provide output data, such as measurement data 124, representing the results of circuit implementation, and transmit this data to classical processor 102. Each data point in the measurement data 124 received from quantum computing hardware 104 can include a bit string representing the measured quantum state of the quantum system after a corresponding random benchmark circuit is applied to the quantum system. For example, for a quantum system comprising two qubits, the measurement data can include multiple sets of data points, each set corresponding to a corresponding random benchmark circuit, and each set including a bit string with values ​​of 00, 01, 10, or 11 (representing quantum states |00>, |01>, |10>, and |11>, respectively). The number of sets of data points is equal to the number of different gate sequences defined by random quantum circuit generator 110 multiplied by the total number of different periods used by random quantum circuit generator 110 to generate quantum circuit data 116. The number of data points in each set is equal to the number of times the corresponding random benchmark circuit is implemented and measured by the quantum computing hardware. This number can be a pre-set system parameter or can be specified by input data 106 and / or quantum circuit data 116.

[0053] Data processing module 114 is configured to process measurement data 124 received from quantum computing hardware to determine quantum state purity, representing the average purity of the quantum state output by quantum hardware 104. For example, data processing module 114 may be configured to perform the following (see reference below). Figure 3 Example process 300 is described.

[0054] Classical processor 102 provides output data 108 representing the determined purity of the quantum state. In some embodiments, classical processor 102 may also be configured to use the determined quantum state purity to determine one or more adjustments to quantum computing hardware 104, such as adjustments to control parameters of a control model used to perform quantum operations. The data 128 representing the determined adjustments may be provided to and implemented by quantum computing hardware 104 when future computations are performed to improve the operation and / or performance of quantum computing hardware 104. As an example, adjustments may be based on an optimization cost function that depends on the purity of the parameters relative to the control model.

[0055] In some implementations, the classical processor 102 may also be configured to perform benchmark experiments to determine the quantum state fidelity realized by the output state of the quantum hardware 104. The determined quantum state fidelity can be used in conjunction with the determined quantum state purity to distinguish the type of error caused by the quantum computing hardware 104 and to determine adjustments to how to control the quantum computing hardware 104, as referenced below. Figure 2 A more detailed description.

[0056] Programming the hardware

[0057] Figure 2 This is a flowchart of an example process 200 for determining the average purity of multiple output quantum states, where the multiple output quantum states correspond to the application of corresponding random quantum circuits of the same circuit depth to the same initial quantum state. For convenience, process 200 will be described as being performed by a system of one or more classical and quantum computing devices located in one or more locations. For example, appropriately programmed according to this specification... Figure 1 System 100 can execute process 200.

[0058] The system obtains measurement data (step 202). The measurement data corresponds to the measurement results of a random quantum circuit of the same circuit depth applied to the same initial quantum state.

[0059] Each data point in the acquired measurement data can include a string of bits representing the measured quantum state of the quantum system after the corresponding random quantum circuit is applied. For example, for a quantum system comprising two qubits, the measurement data can include multiple sets of data points, each corresponding to a corresponding random quantum circuit, and each set of data points includes a string of bits taking the values ​​00, 01, 10, or 11 (representing quantum states |00>, |01>, |10>, and |11>, respectively). The number of sets of data points is equal to the number of random quantum circuits implemented by the quantum computing hardware. The number of data points in each set is equal to the number of times the corresponding random quantum circuit is implemented and measured.

[0060] The system calculates the probability P of applying the corresponding random quantum circuit to the initial quantum state to produce the corresponding measurement result from the measurement data. m (Step 204). For example, for a quantum system comprising two qubits, the system calculates the probability that each random quantum circuit generates a bit string 00, 01, 10, 11 corresponding to quantum states |00>, |01>, |10>, |11>, respectively. Calculating the probability may include dividing the number of times the measurement occurs by the total number of measurements from the random quantum circuit. Example graphical representations of the calculated probabilities of corresponding measurements from multiple random quantum circuits (30 circuits and up to 500 cycles) are shown below. Figure 3 It is shown and described.

[0061] The system calculates the probability statistics of the measurement results to determine the average purity of the multiple output quantum states. The system represents each output quantum state by the depolarization channel given in equation (2) below.

[0062]

[0063] In equation (2), p represents the probability of a pure-state output |ψ>, (1-p) represents the probability that the output state is a fully decoherent state, D represents the size of the corresponding quantum system's Hilbert space, and II represents the identity operator. The depolarization channel given in equation (2) indicates that each output quantum state can be numerically verified or confirmed by analogy to the rotational argument applied in random benchmarks: the quantum graph corresponding to the physical error is randomly unitary conjugated, which gives the depolarization channel.

[0064] Combining equations (1) and (2), the purity of the output state is given by equation (3) below.

[0065] Purity = p 2 (3)

[0066] From equation (2), it can be seen that for p = 0, the probability of each output is equal to 1 / D. In this case, the calculated probability distribution is a delta function located at 1 / D (the integral histogram is a step function, such as...). Figure 3 (As shown). In contrast, if p = 1, the calculated probability P of the measurement result... m Follows the Porter-Thomas distribution.

[0067]

[0068] It has a mean 1 / D and a variance

[0069]

[0070] For any p, the distribution of the calculated probability can be described by the Porter-Thomas distribution of equation (4), which shrinks to the mean 1 / D by a factor p. Since the uniform distribution has no variance, the variance of the calculated probability distribution is equal to p of the Porter-Thomas variance given in equation (5). 2 times.

[0071] Therefore, to determine the average purity, the system calculates the variance of the calculated probabilities (step 206) and divides the variance of the calculated probabilities by the variance of the Porter-Thomas distribution given in equation (5) (step 208). That is, the system determines...

[0072]

[0073] The average purity determined in step 208 can be used as a measure of the purity of each output quantum state (i.e., each random quantum circuit). This is because the average purity is determined using multiple random quantum circuits of the same depth, and therefore with the same number of gates. To determine the average purity corresponding to circuits of multiple different depths, example process 200 can be repeated for each depth. Furthermore, random quantum circuits can be designed such that the corresponding output states have similar purities. For example, as referenced above... Figure 1 To determine the purity of a single qubit, all single-qubit gates included in a random quantum circuit can have similar error rates. For two qubit purities, the random quantum circuit can include the same two-qubit gates, where only the single-qubit gates vary between circuits.

[0074] In some implementations, the system can use the determined average purity to determine one or more adjustments to the quantum computing hardware, such as adjustments to the control parameters of the control model used by the quantum computing hardware to implement quantum operations. These determined adjustments can be implemented during future computations to improve the operation and / or performance of the quantum computing hardware.

[0075] In some implementations, the system can further identify system control errors in the implementation of the random quantum circuit. For example, the system can further process the measurement data obtained in step 202 using techniques such as cross-entropy benchmarking to determine a metric of quantum state fidelity. The system can then compare the determined fidelity with the square root of the average purity determined in step 208 and given by equation (6) to verify their dependence on the number of cycles d (e.g., the depth of the random quantum circuit). In the absence of system control errors, the determined average purity and the square root of the determined quantum state fidelity should be equal. However, experimentally, control errors are often present. The presence of control errors can lead to incorrect predictions of the output state |ψ>, thus introducing a higher error into the determined fidelity than the determined purity. Therefore, the system can determine the fidelity loss and the purity loss for each cycle, and determine the difference between the determined fidelity loss and the purity loss for each cycle to obtain the system control error for each cycle.

[0076] In some implementations, the system can further determine adjustments to the system control parameters to reduce identified system control errors, since the presence of system errors indicates experimental space for improvement in how the system operates. For example, the awareness of the existence of non-negligible control errors can be used to adjust the control model used to implement the corresponding operation (e.g., a quantum gate) to explain different types of interactions. Example adjustments include running more complex optimizations to better learn the control model, or adding additional terms to the control model to explain new interactions. The control model can represent a mapping between the parameters of the quantum gate (e.g., qubit rotation angle, phase, etc.) and the control parameters of the physical system used to implement the quantum gate / circuit (e.g., control line voltage, pulse shape, operating frequency, etc.).

[0077] Figure 3 Three example figures (a), (b), and (c) are shown. Figure (a) plots the probabilities of measurement results from multiple cross-entropy benchmark experiments (i.e., multiple random quantum circuits defined for cross-entropy benchmark experiments). For example, the probabilities plotted in Figure (a) could correspond to the above... Figure 2 The probabilities calculated in step 204. Each point in Figure (a) represents the probability P(|00>), P(|10>), P(|01>), and P(|11>) of the corresponding cross-entropy benchmark experiment generating the output bit strings 00, 01, 10, and 11. The shaded points represent the corresponding probabilities as defined in keyword 302. For example, arrow 304 points to a point that represents the probability that a random cross-entropy benchmark circuit 20 with a depth of 6 (from a depth range of 1 to 500) (from 30 possible random benchmark circuits) generates the output bit string 00. As another example, arrow 306 points to a point that represents the probability that a random cross-entropy benchmark circuit 5 with a depth of 400 (from the same depth range) (from the same 30 possible random benchmark circuits) generates the output bit string 01.

[0078] Figure (a) shows the characteristic speckle patterns at low cycle numbers (e.g., near dashed line 308) for 30 randomized benchmark circuits and probabilities P(|00>), P(|10>), P(|01>), and P(|11>). The speckle contrast decreases with the cycle number as decoherence begins to dominate the dynamics, for example, near dashed line 310.

[0079] Figure (b) is an integral histogram (also known as the cumulative distribution) of the probabilities corresponding to the vertical lines 308 and 310 in Figure (a), i.e., the probabilities P(|00>), P(|10>), P(|01>), and P(|11>) obtained on 30 random benchmark circuits with fixed circuit depths of 12 and 490. These probabilities have been normalized by the inverse of the Hilbert space dimension, so the uniform distribution described below is approximately 1.

[0080] Figure (b) shows how well the probability distribution is described by the Porter-Thomas distribution at low period numbers. This is indicated by arrow 312 corresponding to line 308. Figure (b) also shows how the probability distribution approaches a uniform distribution (the step function in the integral histogram) at high period numbers. This is indicated by arrow 314 corresponding to line 310. The transition from the Porter-Thomas distribution to a uniform distribution is a result of the quantum system being benchmarked being exposed to more decoherence (as the period number increases) and a decrease in state purity.

[0081] Figure (c) plots the square root of purity and cross-entropy benchmark fidelity (on a logarithmic scale) for each circuit depth within the range of 1-500, where the square root of purity has been taken and an exponential fit performed to make purity directly correlated with the cross-entropy benchmark fidelity loss per cycle. Figure (c) shows the variance of the probability distribution directly correlated with the mean state purity. Figure (c) shows three exponential decays corresponding to purity derived according to the techniques described in this specification, purity derived using tomography, and cross-entropy benchmark fidelity. The number 0.00276 per cycle derived from purity (determined by the exponential fit) and the similar number 0.00282 per cycle derived from the tomographic measurement of purity (also determined by the exponential fit) show good agreement—the techniques currently described and tomographic techniques give similar numbers for purity loss per cycle. The error in the cross-entropy benchmark (including control error) is slightly higher, at 0.00349 per cycle.

[0082] The digital and / or quantum themes and implementations of digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuits, suitable quantum circuits, or more generally in quantum computing systems, in tangibly embodied digital and / or quantum computer software or firmware, in digital and / or quantum computer hardware (including the structures disclosed in this specification and their structural equivalents), or in a combination of one or more of these. The term "quantum computing system" may include, but is not limited to, a quantum computer, a quantum information processing system, a quantum cryptography system, or a quantum simulator.

[0083] The embodiments of the digital and / or quantum themes described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., one or more digital and / or quantum computer program instructions encoded on a tangible, non-transitory storage medium for operation by a data processing device or for controlling the operation of a data processing device. The digital and / or quantum computer storage medium can be a machine-readable storage device, a machine-readable storage substrate, a random or serial access memory device, one or more qubits, or a combination thereof. Alternatively or additionally, the program instructions can be encoded on artificially generated propagation signals capable of encoding digital and / or quantum information, such as machine-generated electrical, optical, or electromagnetic signals, generated to encode digital and / or quantum information for transmission to a suitable receiver device for operation by a data processing device.

[0084] The terms quantum information and quantum data refer to information or data carried, stored, or preserved by a quantum system, where the smallest nontrivial system is a qubit, i.e., a system that defines a unit of quantum information. It is understood that the term "qubit" encompasses all quantum systems that can be appropriately approximated as a two-level system in the corresponding context. Such quantum systems can include multi-level systems, e.g., systems with two or more levels. For example, such systems can include atoms, electrons, photons, ions, or superconducting qubits. In many implementations, the fundamental computational states are identified using a ground state and a first excited state; however, it is understood that other arrangements where computational states are identified using higher-level excited states are possible.

[0085] The term "data processing device" refers to digital and / or quantum data processing hardware and encompasses all kinds of devices, apparatuses, and machines for processing digital and / or quantum data, including, for example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The device may also be, or may further include, dedicated logic circuitry such as FPGAs (Field-Programmable Gate Arrays), ASICs (Application-Specific Integrated Circuits), or quantum simulators—quantum data processing devices designed to simulate or generate information about a particular quantum system. In particular, a quantum simulator is a special-purpose quantum computer that does not have the ability to perform general-purpose quantum computing. In addition to hardware, the device may optionally include code that creates the operating environment for digital and / or quantum computer programs, such as code constituting processor firmware, protocol stacks, database management systems, operating systems, or combinations thereof.

[0086] Digital computer programs, also referred to or described as programs, software, software applications, modules, software modules, scripts, or code, can be written in any form of programming language (including compiled or interpreted languages, or declarative or procedural languages), and can be deployed in any form (including as standalone programs or as modules, components, subroutines, or other units suitable for use in a digital computing environment). Quantum computer programs, also referred to or described as programs, software, software applications, modules, software modules, scripts, or code, can be written in any form of programming language (including compiled or interpreted languages, or declarative or procedural languages) and translated into a suitable quantum programming language, or can be written in a quantum programming language (e.g., QCL or Quipper).

[0087] Digital and / or quantum computer programs may, but do not necessarily, correspond to files in a file system. Programs may be stored as a portion of a file containing other programs or data (e.g., in one or more scripts within a markup language document), in a single file dedicated to the relevant program, or in multiple coordinating files (e.g., a file storing one or more modules, subroutines, or portions of code). Digital and / or quantum computer programs may be deployed to run on a single digital computer or a single quantum computer, or on multiple digital and / or quantum computers located at one site or distributed across multiple sites and interconnected via digital and / or quantum data communication networks. A quantum data communication network is understood as a network that can use quantum systems (e.g., qubits) to transmit quantum data. Generally, digital data communication networks cannot transmit quantum data; however, quantum data communication networks can transmit both quantum data and digital data.

[0088] The processes and logic flows described in this specification can be executed by one or more programmable digital and / or quantum computers, which, where appropriate, operate in conjunction with one or more digital and / or quantum processors, running one or more digital and / or quantum computer programs to perform functions by manipulating input digital and quantum data and generating outputs. The processes and logic flows can also be executed by dedicated logic circuitry (e.g., FPGA or ASIC) or quantum simulators, and the apparatus can also be implemented as dedicated logic circuitry (e.g., FPGA or ASIC), or the processes and logic flows can be executed by a combination of dedicated logic circuitry or quantum simulators and one or more programmable digital and / or quantum computers.

[0089] A system that enables one or more digital and / or quantum computers to perform a specific operation or action means that the system has software, firmware, hardware, or a combination thereof installed thereon, which, in operation, causes the system to perform said operation or action. A program that enables one or more digital and / or quantum computers to perform a specific operation or action means that the one or more programs include instructions that, when run by a digital and / or quantum data processing device, cause that device to perform said operation or action. A quantum computer can receive instructions from a digital computer that, when run by a quantum computing device, cause that device to perform said operation or action.

[0090] Digital and / or quantum computers suitable for running digital and / or quantum computer programs can be based on general-purpose or special-purpose digital and / or quantum processors or both, or any other type of central digital and / or quantum computer processing unit. Generally, the central digital and / or quantum processing unit receives instructions and digital and / or quantum data from read-only memory, random access memory, or a quantum system suitable for transmitting quantum data (e.g., photons), or a combination thereof.

[0091] Essential components of a digital and / or quantum computer are a central processing unit (CPU) for executing or running instructions and one or more memory devices for storing instructions and digital and / or quantum data. The CPU and memory may be supplemented by or incorporated into dedicated logic circuitry or a quantum simulator. Generally, a digital and / or quantum computer will also include one or more mass storage devices, either operatively coupled to receive or transfer digital and / or quantum data from or to one or more mass storage devices, or both, such as magnetic disks, magneto-optical disks, optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices.

[0092] Digital and / or quantum computer-readable media suitable for storing digital and / or quantum computer program instructions and digital and / or quantum data include all forms of non-volatile digital and / or quantum memories, media, and memory devices, including, for example: semiconductor memory devices such as EPROM, EEPROM, and flash memory devices; disks such as internal hard disks or removable disks; magneto-optical disks; CD-ROMs and DVD-ROMs; and quantum systems such as trapped atoms or electrons. It is understood that quantum memory is a device capable of storing quantum data for long periods with high fidelity and efficiency, such as a light-matter interface where light is used for transmission and matter for storing and preserving quantum features such as superposition or quantum coherence of the quantum data.

[0093] Control of the various systems or portions thereof described in this specification can be implemented in a digital and / or quantum computer program product comprising instructions stored on one or more non-transitory machine-readable storage media and executable on one or more digital and / or quantum processing devices. The systems or portions thereof described in this specification can be implemented as apparatus, methods, or systems comprising one or more digital and / or quantum processing devices and memory for storing operable instructions to perform the operations described in this specification.

[0094] While this specification contains numerous specific implementation details, these details should not be construed as limiting the scope of possible claims, but rather as descriptions of features that may be specific to particular embodiments. Some features described in the context of individual embodiments may also be implemented in combinations of individual embodiments. Conversely, various features described in the context of individual embodiments may also be implemented individually or in any suitable sub-combination in multiple embodiments. Furthermore, although features may be described above as functioning in certain combinations and even initially claimed in this way, in some cases one or more features from a claimed combination may be removed from that combination, and the claimed combination may be for sub-combinations or variations thereof.

[0095] Similarly, although the operations are depicted in a specific order in the accompanying drawings, this should not be construed as requiring these operations to be performed in the specific order shown or sequentially, or to perform all the shown operations to achieve the desired result. In some cases, multitasking and parallel processing can be advantageous. Furthermore, the separation of various system modules and components in the above embodiments should not be construed as requiring such separation in all embodiments; it should be understood that the described program components and systems can generally be integrated together into a single software product or packaged into multiple software products.

[0096] Specific embodiments of the subject matter have been described. Other embodiments are within the scope of the appended claims. For example, the actions recited in the claims can be performed in a different order and still achieve the desired result. As an example, the processes depicted in the drawings do not necessarily require the specific order or sequential order shown to achieve the desired result. In some cases, multitasking and parallel processing can be advantageous.< / z> < / y> < / x>

Claims

1. A method for determining the average purity of a plurality of output quantum states, wherein the plurality of output quantum states correspond to the application of corresponding random quantum circuits of the same circuit depth to the same initial quantum state, the method comprising: Multiple data items are obtained, each data item corresponding to a corresponding random quantum circuit of the same circuit depth, and representing the probability of applying the corresponding random quantum circuit to the initial quantum state to produce the corresponding measurement result; Calculate the variance of multiple data items; Determine a Porter-Thomas distribution with the same dimension as each output quantum state; as well as Divide the calculated variance by the variance of the Porter-Thomas distribution to determine the average purity. Each output quantum state is represented by a depolarization channel with a depolarization channel parameter p, where p represents the probability that the output quantum state is a pure state output. The depolarization channel parameter p is equal to 1, and the multiple data terms are distributed according to a Porter-Thomas distribution. The depolarization channel parameter p is equal to 0, and the plurality of data items are distributed according to a δ function located at 1 / D, where D represents the dimension of the Porter-Thomas distribution.

2. The method according to claim 1, wherein the variance of the Porter-Thomas distribution is equal to Where D represents the dimension of the Porter-Thomas distribution.

3. The method of claim 1, wherein the random quantum circuit comprises a random quantum circuit generated for a cross-entropy benchmark experiment.

4. The method of claim 1, wherein the random quantum circuit comprises a quantum circuit including one or more quantum gates randomly sampled from a predetermined set of quantum gates.

5. The method of claim 1, wherein each random quantum circuit comprises the same number of quantum gates.

6. The method of claim 1, wherein purity includes single-qubit purity, and wherein each random quantum circuit comprises a plurality of single-qubit quantum gates having an error rate within the same predetermined range.

7. The method of claim 1, wherein purity includes n-qubit purity, and wherein each random quantum circuit includes i) a plurality of single-qubit quantum gates having an error rate within the same predetermined range, and ii) the same n-qubit quantum gate.

8. The method according to claim 1, further comprising: Obtain measurement data that corresponds to the measurement results of applying a corresponding random quantum circuit of the same circuit depth to the same initial quantum state; as well as The obtained measurement data is used to calculate the probability of applying the corresponding random quantum circuit to the initial quantum state to produce the corresponding measurement result.

9. The method according to claim 1, further comprising: The obtained data items are processed to determine the quantum state fidelity of the multiple output quantum states; as well as Calculate the difference between the determined quantum state fidelity and the determined average purity loss, where the calculated difference represents the system control error.

10. The method of claim 9, further comprising: One or more adjustments to the quantum hardware control parameters are determined based on the calculated difference representing the system control error; as well as Implement one or more of the determined adjustments to perform quantum computing using quantum computing hardware.

11. The method according to claim 1, further comprising: One or more adjustments to the control parameters of the quantum hardware are determined based on a defined average purity. as well as Implement one or more of the determined adjustments to perform quantum computing using quantum computing hardware.

12. An apparatus comprising one or more classical and / or quantum storage devices storing operable instructions that, when executed by one or more computers, cause one or more computing devices to perform operations, said operations comprising the method according to any one of claims 1 to 11.

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