Characterization of Entangled Time-Correlated Quantum Errors

Abstract

Errors affecting a quantum computer can be effectively measured and characterized by placing the quantum computer in a highly entangled state such as a Greenberger-Horne-Zeilinger (GHZ) state, accumulating quantum errors in the highly entangled state, and then measuring the accumulated errors. In some approaches, error characterization includes measuring the parity oscillations of the GHZ state and fitting a quantum error model to the power spectrum of the parity oscillations. The fitted quantum error model can be used to select a suitable error-tolerant error correction scheme for the quantum computer given the ambient noise.

Description

[0001] Cross - Reference to Related Applications

[0002] This application claims the benefit of U.S. Provisional Application No. 63 / 032,459, filed on May 29, 2020, the content of which is incorporated herein by reference. Background Art

[0003] Quantum computers may be able to perform computations that classical computers cannot, such as efficient prime factorization. At the same time, environmental noise that affects the fidelity of the quantum states of a quantum computer degrades the performance of the quantum computer. Summary of the Invention

[0004] The performance of a quantum computer is limited by time - dependent or time - reliant quantum errors, such as time - varying quantum error characteristics (such as 1 / f noise or quantum gate control drift). Measuring these time - dependent quantum error characteristics provides an understanding of the viable range of quantum computing and informs decisions regarding error - tolerant correction schemes given the system noise.

[0005] The subject matter of this disclosure provides a technique for measuring quantum error characteristics by using highly entangled quantum states of a quantum computer that are jointly sensitive to quantum errors. For example, N qubits of a quantum computer can be initialized to a Greenberger - Horne - Zeilinger (GHZ) state (i.e., the so - called "Schrödinger cat state"). When such a highly entangled state evolves over time, there is an N - fold amplification of quantum errors through spatial entanglement. Thus, less time is required to measure and characterize quantum errors, and more accurate error diagnosis can be achieved with the same amount of calibration time. The measured quantum errors can be fitted to a quantum error model, such as a model that characterizes random digital phase errors and bit - flip errors, such as the Pauli error model. This efficient measurement of quantum errors enables high - time - resolution characterization of drift and 1 / f noise, as well as a rapid assessment of the applicability and fault tolerance of a quantum computer.

[0006] Aspects of the subject matter of this disclosure include a method of operating a quantum computer, the method including: initializing N qubits of the quantum computer to a highly entangled state (e.g., a Greenberger - Horne - Zeilinger (GHZ) state) selected to amplify quantum errors; accumulating quantum errors in the highly entangled state; and measuring the accumulated errors.

[0007] In some approaches, the method includes repeating a sequence of initialization, accumulation, and measurement; and obtaining an ensemble average of the cumulative errors of the repeated measurements. The repeated measurement sequence can include a sequence of measuring the parity oscillations of a highly entangled state of the phase angle φ sequence, such as an incremental sequence of the phase angle φ or a random sequence of the phase angle φ. Each measurement can include applying an analysis pulse of the phase angle φ to each of the N qubits, and can further include applying a series of dynamic decoupling pulses (such as a CPMG pulse series) after each analysis pulse to mitigate dominant errors.

[0008] In some approaches, these methods include fitting a quantum error model, such as a Pauli error model, to the ensemble average of the cumulative errors. For example, these methods can include obtaining the power spectrum of the parity oscillations according to the phase frequency ω conjugate to the phase angle φ φ and then fitting the Pauli phase error probability ρ z and the Pauli bit flip error probability ρ x to the power spectrum of the parity oscillations.

[0009] In some approaches, the method includes selecting a quantum error correction scheme corresponding to the fitted quantum error model. The method can also include performing a computational process by a quantum computer using the selected quantum error correction scheme. Additionally or alternatively, the method can include evaluating the suitability of the quantum computer for the selected computational process based on the fitted quantum error model.

[0010] Other aspects of the subject matter of this disclosure include a system for operating a quantum computer, wherein the system includes: one or more processors; one or more I / O devices coupled to the one or more processors and configured to send control signals to and receive readout signals from the quantum computer; and one or more memories storing computer-readable instructions thereon, the instructions being configured to cause the one or more processors and the one or more I / O devices to perform any of the above methods on the quantum computer.

[0011] Other aspects of the subject matter of this disclosure include transient and non-transient computer-readable media storing instructions that are executed by one or more computers configured to send control signals to and receive readout signals from a quantum computer, the instructions causing the one or more computers to perform any of the above methods on the quantum computer.

[0012] The various features and advantages of the foregoing subject matter are described below with reference to the accompanying drawings. Additional features and advantages are apparent from the subject matter and claims described herein. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1 is a schematic diagram of an exemplary quantum computer system.

[0014] Figure 2 is a flow chart showing an example of a process of characterizing quantum errors using highly entangled states of a quantum computer.

[0015] Figure 3 is an example of the preparation and measurement of a highly entangled quantum state of a quantum computer.

[0016] Figure 4 depicts an exemplary power spectrum of the parity oscillation of a highly entangled quantum state of a quantum computer.

[0017] Figure 5 depicts an exemplary fit of a quantum error model to the power spectrum of the parity oscillation of a highly entangled quantum state of a quantum computer.

[0018] Like reference numerals and names in the various figures indicate like elements. DETAILED DESCRIPTION

[0019] Time-dependent quantum gate errors can be detrimental to large-scale quantum computing based on error-corrected tolerance and may degrade the performance of most quantum algorithms. The present subject matter provides methods, systems, and computer programs for achieving efficient measurement and characterization of time-dependent gate errors by using quantum entanglement across many qubits.

[0020] Typically, measurement accuracy scales with time and space. In the case of time-dependent signals, scaling the measurement with time may not be desirable because the longer each measurement takes, the lower the maximum measurable frequency becomes (by the Nyquist sampling theorem). Therefore, it is appropriate to increase the measurement accuracy / efficiency by using more sensitive probes.

[0021] Highly entangled states such as the Greenberger-Horne-Zeilinger (GHZ) state (the so-called "Schrödinger's cat" state) provide high sensitivity to collective errors such as phase shifts. GHZ states are commonly used in quantum sensing and metrology, but in these fields, the sensitivity of these states to environmental noise is considered a drawback because the sensitivity can disrupt the measurement of a specific non-environmental signal measured by quantum sensing / metrology. On the other hand, in the present context, the sensitivity of highly entangled states such as GHZ states provides a more sensitive and effective measurement of environmental noise for a quantum computer system. Other examples of highly entangled states include GHZ-like states, cluster states, and cluster-like states. Although the following description involves the use of GHZ states, it should be understood that this can alternatively apply to other highly entangled states.

[0022] Now refer to Figure 1, depicts a block diagram of a quantum computer system 100. This is a schematic description of the general characteristics of a quantum computer system and is not intended to be limiting. Generally, the quantum computer system 100 includes a qubit component 110, and the qubit component 110 includes a set of qubits 112. For example, each qubit can be a two-level quantum system having levels representing logical values 0 and 1; examples of these two-level quantum systems include superconducting transmon states in superconducting quantum computers, hyperfine atomic states in atomic quantum computers, and nuclear spin states in NMR quantum computers. Although Figure 1 depicts the qubits 112 as arranged in a rectangular array, this is a schematic description and is not intended to be limiting.

[0023] The qubit component 110 typically also includes adjustable coupling elements 114 to allow quantum gate operations between the coupled qubits. In Figure 1 the schematic depiction, each qubit 112 is adjustably coupled to each of its four adjacent qubits through these couplers 114. Again, this is a schematic arrangement of qubits and couplers, and other arrangements are possible, including non-rectangular arrangements, arrangements allowing coupling between non-adjacent qubits, and arrangements including adjustable coupling between more than two qubits.

[0024] Generally, a quantum processing algorithm is performed by initializing the qubits in selected initial states and then applying a sequence of quantum gate operations to the qubits. The sequence of quantum gate operations can include: single-qubit gate operations such as X, Y, Z, or Hadamard gates; two-qubit gate operations such as CX, CZ, and SWAP; and gate operations involving three or more qubits such as Toffoli gates. The gate operations can be performed by applying various control signals 132 to the qubits 112 and the couplers 114; these can include, for example, RF or microwave pulses in NMR or superconducting quantum computer systems, or optical pulses in atomic quantum computer systems. At the end of the quantum processing algorithm, the final state of the qubits can be measured using a quantum observable such as Z with a readout signal 134, where the readout signal 134 can again be an RF, microwave, or optical signal depending on the physical scheme used for the quantum computer. In Figure 1 for simplicity, the control signals and the readout signal are depicted as only addressing selected elements of the qubit component (i.e., the top and bottom rows), but these control and readout signals can address each element in the qubit component.

[0025] A quantum computer system 100 generally includes a control and measurement system 120, which may include one or more classical (i.e., non-quantum) processors 122, one or more memories 124, and one or more I / O units 126 connected by one or more data buses 126. The control and measurement system can be programmed to send a sequence of selected control signals 132 to a qubit component (e.g., to perform a selected series of quantum gate operations) and receive a sequence of readout signals 134 from the qubit component (e.g., to perform a selected series of measurements). Thus, for example, one or more I / O units 126 may include D / A converters, A / D converters, and RF / microwave / optical signal generators, transmitters, and receivers depending on the physical scheme of the quantum computer.

[0026] Now referring to Figure 2 , a flowchart is depicted that illustrates an example of a process 200 for characterizing quantum errors using a highly entangled state of a quantum computer system. Process 200 includes step 210 - initializing N qubits of the quantum computer in a highly entangled state. For example, using the Figure 1 quantum computer system 100, control signals 132 can be used to initialize N qubits of the qubit component 110 in a highly entangled state. The N qubits can constitute all the qubits in the qubit component 110, or it can constitute a selected subset of qubits in the qubit component. In some means, the highly entangled state is a Greenberger-Horne-Zeilinger (GHZ) state, i.e., an equal superposition of all qubits in state 0 and all qubits in state 1. For an example where N = 5, a GHZ state can be prepared using, for example, a series of Hadamard and controlled-NOT (CX) gates shown at circuit level 301 in the Figure 3 quantum circuit. Other suitable sequences of one- and two-qubit gates can be used to prepare the GHZ state, depending on the gates available for the physical scheme of the quantum computer.

[0027] Process 200 further includes step 212 - accumulating quantum errors in the highly entangled state. This can include, for example, waiting for a period of time after the final preparation of the highly entangled state before starting to measure the qubits so that quantum errors due to environmental noise can accumulate. Alternatively or additionally, when the quantum computer is initialized in a highly entangled state, quantum errors due to environmental noise accumulate, for example, during a series of gate operations (such as the Hadamard and CX gate operations 301 for preparing the GHZ state, as shown in Figure 3 ) for preparing the highly entangled state.

[0028] Process 200 also includes step 214 - measuring the accumulated errors. For example, using the Figure 1A quantum computer system 100 can collect readout signals 134 to measure the state of each qubit in a highly entangled state to observe the accumulated environmental noise.

[0029] In one approach, the measurement of the accumulated error includes measuring the parity oscillation of the quantum computer. For example, after the system has been prepared in a GHZ state using, e.g., Figure 3 the Hadamard and CX gates 301, the measurement can be performed by applying an analysis pulse P j (φ) to each qubit in the prepared GHZ state (as shown at the circuit level 302 of Figure 3 ) and then measuring the Z observable of each qubit (as shown at the circuit level 303 of Figure 3 ). Each analysis pulse is a Bloch rotation, which can be expressed in the form of , where X j and Y j are the Pauli operators of the j-th qubit, and φ is the selected phase angle for the parity oscillation measurement. This is equivalent to measuring the following expectation value for 0 < φ < 2Π, where φ may or may not be related to t:

[0030]

[0031] where ρ is the density matrix of the quantum bit quantum ensemble that has actually been prepared (representing the pure GHZ state with added quantum errors and environmental noise), is the analysis pulse with the phase angle φ applied to all qubits, and is the tensor product of the Pauli Z matrices of all qubits. In some approaches, a series of dynamic decoupling pulses such as the Carr-Purcell-Meiboom-Gill (CPMG) pulse series can be applied after the analysis pulse to improve the measurement by correcting the phase shift error.

[0032] The steps of initialization 210, accumulation of quantum error 212, and measurement 214 can be repeated sequentially multiple times to increase the statistical sample size. For example, if each measurement step is the measurement of the parity oscillation for a selected phase angle φ as described above, the initialization, accumulation, and measurement can be repeated multiple times for the selected phase angle φ, and / or the initialization, accumulation, and measurement can be repeated for multiple values of the phase angle φ in the detection phase angle range 0 < φ < 2π. In one approach, the initialization, accumulation, and measurement are repeated for a phase sequence that is an increasing sequence such that φ increases with time during the repetition (i.e., the phase measurements are time-ordered). In another approach, the initialization, accumulation, and measurement are repeated for a random phase sequence, so φ is not time-ordered.

[0033] For means where initialization, accumulation, and measurement are repeated, process 200 also includes step 220 of obtaining an ensemble average of the accumulated errors. Thus, repeated measurements are compiled to increase the statistical sample size, and for means involving parity oscillation measurements, the parity oscillations are measured over a phase angle range of 0 < φ < 2π.

[0034] Obtaining the ensemble average can include obtaining the power spectrum of the parity oscillations according to the phase frequency ω conjugate to the phase angle φ φ The power spectrum can reveal specific characteristics of quantum errors (such as phase errors and bit flip errors) that affect the quantum computer. Considering a pure GHZ state unaffected by environmental noise, it has a power spectrum according to the phase frequency ω φ which can be represented as follows:

[0035]

[0036] In other words, for the GHZ state of N entangled qubits, the parity oscillations of the pure GHZ state are a cosine wave with an amplitude of 1 and a frequency of N.

[0037] In the presence of environmental noise introducing quantum errors such as phase errors and bit flip errors, the phase error contributes to the main mode at frequency ω φ = N by suppressing the main mode

[0038]

[0039] (where Z j represents the effect of the phase error on qubit j); while the bit flip error results in lower frequency modes at frequencies ω φ = N - 2, N - 4, etc.

[0040]

[0041] (where X j represents the effect of the bit flip error on qubit j). The cumulative result is that the power spectrum of the parity oscillations of the GHZ state can have the following form, revealing the characteristic signatures of both phase errors and bit flip errors:

[0042]

[0043] where the coefficient α N of the main mode reveals the phase suppression error rate of the quantum computer system, and the coefficients α N-2 , α N-4 etc. of the low frequency modes reveal the bit flip error rate of the quantum computer system.

[0044] Now refer to Figure 4, presenting the experimental data of the parity oscillation in the GHZ state of 12 qubits. The four panels depict the power spectra of the parity oscillation according to the phase frequency ω φ for the phase time ordering with and without the analysis pulse (as described above) and the experiments with and without the CPMG pulse sequence (as described above) after each analysis pulse. Each panel shows the main parity oscillation modes at ω φ = ±N (in this case = ±13). Using scrambled phases can serve to whiten the time-correlated errors, while using the CPMG pulse sequence can serve to correct the phase shift errors. Therefore, the similarity between the two power spectra collected with the CPMG pulse sequence indicates that most of the differences between the two non-CPMG experiments can be attributed to the phase shift errors.

[0045] Return Figure 2 , process 200 also includes step 230 - fitting a quantum error model to the ensemble average of the accumulated errors. For example, the quantum errors accumulated in the GHZ state can be described by a digital error model (such as the Pauli error model), in which the environmental noise affecting the qubit components is modeled as the random insertion of single-qubit Z gates (resulting in phase errors) and / or the random insertion of single-qubit X or Y gates (resulting in bit flip errors). As described above, the parameters of such a model (e.g., the phase error probability ρ z and the bit flip error probability ρ x ) are reflected in the power spectra of the parity oscillation of the GHZ state. Therefore, by fitting the model to the power spectra of the parity oscillation, a quantum error model such as the Pauli error model can be fitted to the ensemble average of the accumulated errors.

[0046] Figure 5 Presents two examples of such fits to the power spectra obtained from the experimental parity oscillation of the GHZ state. On the left, the power spectrum of the parity oscillation of the N = 16 GHZ state is fitted to the Pauli error model by the least squares method, where the error ratio of Pauli Z to Pauli X is 4.8. On the right, the power spectrum of the parity oscillation of the N = 12 GHZ state is fitted to the Pauli error model by the least squares method, where the error ratio of Pauli Z to Pauli X is 6.3.

[0047] Figure 2 Process 200 also includes step 240 - selecting a quantum error correction scheme corresponding to the fitted quantum error model. For example, based on the fitting parameters of the fitted quantum error model, the redundancy of the error-tolerant error correction scheme can be determined, such as the distance of the error correction code can be determined, and then the quantum algorithm can be modified to use the error-tolerant error correction scheme with the determined redundancy.

[0048] Figure 2 The process 200 may further include step 250 - performing a quantum computing process using a quantum error correction scheme. For example, in the case where an error-tolerant error correction scheme corresponding to a fitted quantum error model with an appropriate redundancy for the computing task at hand has been found, the determined appropriate redundancy can be used to perform the quantum computing process.

[0049] Return Figure 1 , the control and measurement system 120 provides an example of a classical (i.e., non-quantum) computer system that can be used to perform various operations on the qubit components 110, as described above. System 120 includes a processor 122, a memory 124, and an input / output device 126. Each of the components 122, 124, and 126 can be interconnected, for example, using a system bus 128. The processor 122 is capable of processing instructions for execution within the system 120. In some embodiments, the processor 122 is a single-threaded processor. In other embodiments, the processor 122 is a multi-threaded processor. The processor 122 is capable of processing instructions stored in the memory 124.

[0050] The memory 124 stores information within the system 120. In some embodiments, the memory 124 includes a computer-readable medium, volatile storage units, and / or non-volatile storage units. In some means, the memory 124 can include a storage device capable of providing mass storage for the system 120, such as a hard disk device, an optical disc device, a storage device shared by multiple computing devices (e.g., a cloud storage device) over a network, and / or some other mass storage device.

[0051] The input / output device 126 provides input / output operations for the system 120. As described above, the input / output device can include a D / A converter, an A / D converter, and RF / microwave / optical signal generators, transmitters, and receivers to appropriately send control signals 132 to and receive readout signals 134 from the qubit components according to the physical scheme of the quantum computer. In some embodiments, the input / output device 126 may further include one or more network interface devices, such as an Ethernet card, a serial communication device (e.g., an RS-232 port), and / or a wireless interface device (e.g., an 802.11 card). In some embodiments, the input / output device can include a drive device configured to receive input data and send output data to other external devices, such as a keyboard, a printer, and a display device.

[0052] Although already in Figure 1An example control and measurement system 120 is depicted, but embodiments of the subject matter and functional operations described in this specification can be implemented in other types of digital electronic circuitry or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them.

[0053] Embodiments of the subject matter and operations described in this specification can be implemented in digital electronic circuitry or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them. Embodiments of the subject matter described in this specification can be implemented as one or more computer programs, i.e., one or more circuits of computer program instructions encoded on one or more computer storage media for execution by, or to control the operation of, a data processing apparatus. Optionally or additionally, the program instructions can be encoded on an artificially generated propagated signal (e.g., a machine-generated electrical, optical, or electromagnetic signal) that is generated to encode information for transmission to a suitable receiver apparatus for execution by the data processing apparatus. A computer storage media can be or include a computer-readable storage device, a computer-readable storage substrate, a random or serial access memory array or device, or a combination of one or more of them. Additionally, although a computer storage media is not a propagated signal, a computer storage media can be a source or destination of computer program instructions encoded in an artificially generated propagated signal. The computer storage media can also be or include one or more separate physical components or media (e.g., multiple CDs, disks, or other storage devices).

[0054] The operations described in this specification can be implemented as operations performed by a data processing apparatus on data stored on one or more computer-readable storage devices or received from other sources.

[0055] The term “data processing apparatus” encompasses all kinds of apparatus, devices, and machines for processing data, including, by way of example, a programmable processor, a computer, a system on a chip, or multiple or combinations of the foregoing. The apparatus can include dedicated logic circuitry, such as an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit). In addition to hardware, the apparatus can also include code that creates an execution environment for the computer programs involved, such as code that constitutes processor firmware, a protocol stack, a database management system, an operating system, a cross-platform runtime environment, a virtual machine, or a combination of one or more of them. The apparatus and the execution environment can implement various different computing model infrastructures, such as web services, distributed computing, and grid computing infrastructures.

[0056] A computer program (also known as a program, software, software application, script, or code) can be written in any form of programming language, including compiled or interpreted languages, declarative or procedural languages, and it can be deployed in any form, including as a stand-alone program or as a module, component, subroutine, object, or other unit suitable for use in a computing environment. A computer program may or may not correspond to a file in a file system. The program can be stored as part of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple cooperating files (e.g., files that store one or more modules, subroutines, or portions of code). A computer program can be deployed to execute on one computer or on multiple computers located at one site or distributed across multiple sites and interconnected by a communication network.

[0057] The processes and logical flows described in this specification can be performed by one or more programmable processors executing one or more computer programs to perform actions by operating on input data and generating output. The processes and logical flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, such as an FPGA (Field Programmable Gate Array) or an ASIC (Application Specific Integrated Circuit).

[0058] Processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors. In general, a processor will receive instructions and data from a read only memory or a random access memory or both. The essential elements of a computer are a processor for performing actions in accordance with the instructions and one or more memory devices for storing the instructions and data. In general, a computer will also include one or more mass storage devices for storing data, such as magnetic disks, magneto-optical disks, or optical disks, or operatively coupled to receive data from or transfer data to the one or more mass storage devices. However, a computer need not have such devices. In addition, a computer may be embedded in another device, such as a mobile telephone, a personal digital assistant (PDA), a mobile audio or video player, a game console, a Global Positioning System (GPS) receiver, or a portable storage device (e.g., a Universal Serial Bus (USB) flash drive), to name just a few. Devices suitable for storing computer program instructions and data include all forms of non-volatile memory, media, and memory devices, including by way of example semiconductor memory devices, such as EPROM, EEPROM, and flash memory devices; magnetic disks, such as internal hard disks or removable disks; magneto-optical disks; and CD-ROM and DVD-ROM disks. The processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry.

[0059] To provide for interaction with a user, embodiments of the subject matter described in this specification may be implemented on a computer having: a display device, such as a CRT (cathode ray tube) or LCD (liquid crystal display) monitor, for displaying information to the user; and a keyboard and a pointing device, such as a mouse or a trackball, by which the user may provide input to the computer. Other kinds of devices may also be used to provide for interaction with the user; for example, feedback provided to the user may be any form of sensory feedback, such as visual feedback, auditory feedback, or tactile feedback; and input received from the user may be in any form, including sound, speech, or tactile input. Additionally, the computer may interact with the user by sending documents to or receiving documents from the device used by the user; for example, sending a web page to a web browser on a client device of the user in response to a request received from the web browser.

[0060] Embodiments of the subject matter described in this specification may be implemented in a computing system that includes a back-end component, such as a data server; or includes a middleware component, such as an application server; or includes a front-end component, such as a client computer having a graphical user interface or a web browser, by which the user may interact with an implementation of the subject matter described in this specification; or includes any combination of one or more such back-end, middleware, or front-end components. The components of the system may be interconnected by any form or medium of digital data communication - for example, a communication network. Examples of communication networks include local area networks ("LANs") and wide area networks ("WANs"), the Internet (e.g., the Internet), and peer-to-peer networks (e.g., ad hoc peer-to-peer networks).

[0061] The computing system may include clients and servers. The clients and servers are typically remote from each other and typically interact via a communication network. The relationship of client and server arises by virtue of computer programs running on the respective computers and having a client-server relationship to each other. In some embodiments, the server sends data (e.g., an HTML page) to the client device (e.g., for purposes of displaying data to and receiving user input from a user interacting with the client device). Data generated at the client device (e.g., results of user interaction) may be received at the server.

[0062] Although this specification contains many specific implementation details, these should not be construed as limiting the scope of any invention or of what is claimed, but rather as descriptions of features specific to particular embodiments of a particular invention. Certain features that are described in this specification in the context of separate embodiments can also be implemented in combination within a single embodiment. Conversely, the various features that are described in the context of a single embodiment can also be implemented separately or in any suitable sub-combination in multiple embodiments. Moreover, although the features may be described above as acting in certain combinations and even initially claimed as such, one or more features from a claimed combination can in some cases be removed from the combination, and the claimed combination can be directed to a sub-combination or variation of a sub-combination.

[0063] Similarly, although operations are depicted in the drawings in a particular order, this should not be understood as requiring that the operations be performed in the particular order shown or in sequential order, or that all of the illustrated operations be performed, to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous. Moreover, the separation of various system components in the foregoing embodiments should not be understood as requiring such separation in all embodiments, and it should be understood that the described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0064] Accordingly, particular embodiments of the subject matter have been described. Other embodiments are within the scope of the appended claims. In some cases, the actions recited in the claims can be performed in a different order and still achieve the desired result. Additionally, the processes depicted in the figures do not necessarily require the particular order shown or sequential order to achieve the desired result. In certain implementations, multitasking and parallel processing may be advantageous.

Claims

1. A method of operating a quantum computer, comprising: For each phase angle φ in a sequence of phase angles φ: Initializing N qubits of the quantum computer in a GHZ (Greenberger-Horne-Zeilinger) state, the GHZ state allowing amplification of quantum errors through coherent phase accumulation; Accumulating quantum errors in the GHZ state; and Measuring the parity oscillation of the GHZ state of the phase angle φ; The ensemble average of the accumulated quantum errors obtained by one or more classical computers from the repeated measurements, including according to the phase frequency ω conjugate to the phase angle φ φ Obtain the power spectrum of the parity oscillation; And Fitting, by the one or more classical computers, a Pauli error model to the ensemble average of the accumulated quantum errors, including fitting a Pauli phase error probability Ψ z and a Pauli bit flip error probability Ψ x to the power spectrum of the parity oscillations.

2. The method according to claim 1, wherein The sequence of phase angles φ is an increasing sequence of phase angles φ.

3. The method according to claim 1, wherein The sequence of phase angles φ is a randomized sequence of phase angles φ.

4. The method according to claim 1, wherein For each phase angle φ in the sequence of phase angles φ, measuring the parity oscillation of the GHZ state of the phase angle φ includes: Apply the Bloch rotation of the phase angle φ to each of the N qubits, and the Bloch rotation P j (φ) is of the form, where X j and Y j are the Pauli operators of the j-th qubit; and Measuring the Z observable of each of the N qubits.

5. The method according to claim 4, wherein For each phase angle φ in the sequence of phase angles φ, measuring the parity oscillation of the GHZ state of the phase angle φ includes: Correcting the phase shift error by applying a series of dynamic decoupling pulses after applying the Bloch rotation and before measuring the Z observable of each of the N qubits.

6. The method according to claim 5, wherein, Each of the series of dynamic decoupling pulses is a Carr-Purcell-Meiboom-Gill (CPMG) pulse series.

7. The method according to claim 1, further comprising: Selecting a quantum error correction scheme corresponding to the fitted quantum error model based on the fitting parameters of the fitted Pauli error model.

8. The method according to claim 7, further comprising: Performing a computational process by the quantum computer using the selected quantum error correction scheme.

9. The method according to any one of claims 1-8, further comprising: Evaluating the suitability of the quantum computer for a selected computational process by the one or more classical computers based on the fitted Pauli error model.

10. A system for operating a quantum computer, the system comprising: One or more classical processors; One or more I / O devices, the one or more I / O devices being coupled to the one or more classical processors and configured to send control signals to and receive readout signals from the quantum computer; And One or more memories storing computer-readable instructions, the computer-readable instructions when executed by the one or more classical processors cause the one or more classical processors and the one or more I / O devices to perform the method according to any one of claims 1-9.

11. The system according to claim 10, wherein, The quantum computer is a superconducting quantum computer, an NMR quantum computer, a trapped ion quantum computer, or an atomic quantum computer.

12. The system according to claim 10 or 11, further comprising the quantum computer.

13. A non-transitory computer-readable medium storing instructions, the instructions when executed by one or more classical processors cause the one or more classical processors and one or more I / O devices to perform the method according to any one of claims 1-9.

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