JP7655966B2 - Fidelity Estimation for Quantum Computing Systems - Google Patents

Description

This specification relates to quantum computing.

A quantum circuit is a model of quantum computation in which a computation involves a sequence of quantum gates. Quantum circuits are sensitive to errors due to, for example, decoherence and other quantum noise. The effect of errors in a quantum circuit is sometimes characterized by the fidelity of the quantum circuit. Fidelity is a quantum circuit metric that indicates the quality and reliability of the quantum circuit.

This specification relates to estimating the fidelity of quantum hardware in quantum computing systems. In particular, this specification describes techniques for estimating the fidelity of complex non-Clifford quantum circuits having multiple qubits.

In general, an inventive aspect of the subject matter described herein may be implemented in a method that includes the actions of accessing a set of quantum gates, sampling a subset of quantum gates from the set of quantum gates, where the subset of quantum gates defines a quantum circuit, applying the quantum circuit to a quantum system and performing measurements on the quantum system to determine output information of the quantum system, calculating output information of the quantum system based on application of the quantum circuit to the quantum system, and estimating fidelity of the quantum circuit based on the determined and calculated output information of the quantum system.

Other implementations of the present aspect include corresponding computer systems, devices, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the present method. One or more computer systems can be configured to perform a particular operation or action by installing software, firmware, hardware, or a combination thereof on the system that, when operated, causes the system to perform the action. One or more computer programs can be configured to perform a particular operation or action by including instructions that, when executed by a data processing device, cause the device to perform the action.

These and other implementations may each optionally include one or more of the following features, alone or in combination: In some implementations, estimating the fidelity of the quantum system includes fitting determined output information of the quantum system to calculated output information of the quantum system.

In some cases, fitting the determined output information of the quantum system to the calculated output information of the quantum system to estimate the fidelity of the quantum circuit includes defining a convex combination of the calculated output information of the quantum system and a totally mixed quantum state, and estimating the fidelity of the quantum circuit by comparing the defined convex combination to the determined output information of the quantum system.

In some implementations, the convex combination is given by:

where α represents the fidelity of the quantum circuit, |ψ〉 represents the computed quantum state of the quantum system based on the application of the quantum circuit to the quantum system, and I/N represents the fully mixed state.

In some cases, the method further includes repeatedly sampling subsets of quantum gates from the set of quantum gates until completion of the event, each subset of quantum gates defining a respective quantum circuit; for each sampled subset of quantum gates, applying the respective quantum circuit to a quantum system and performing a respective measurement on the quantum system to determine output information of the quantum system; calculating output information of the quantum system based on application of the respective quantum circuit to the quantum system; and estimating fidelity of the respective quantum circuit based on the determined output information and the calculated output information of the quantum system.

In some implementations, completion of the event occurs when the uncertainty of the estimated fidelity falls below a predefined threshold.

In some cases, the set of quantum gates includes a universal set of quantum gates.

In some cases, the set of quantum gates includes single-qubit quantum gates and two-qubit quantum gates.

In some implementations, each gate in a set of quantum gates is associated with a respective quantum gate fidelity.

In other implementations, the sampled subset of quantum gates includes an equal number of quantum gates of comparable respective quantum gate fidelity.

In some cases, sampling the subset of quantum gates from the set of quantum gates includes randomly sampling the subset of quantum gates from the set of quantum gates.

The subject matter described herein can be implemented in a particular manner to achieve one or more of the following advantages:

Quantum hardware, e.g., systems of quantum gates, are inherently error-prone and errors need to be characterized before they can be corrected. Full characterization by processes such as quantum process tomography is impractical, e.g., in terms of computational cost and efficiency. For example, quantum process tomography becomes prohibitively expensive as the number of qubits in the quantum system increases, as the number of measurements required grows exponentially. In addition, full characterization is often unnecessary, as in practice it may be sufficient to estimate more general quantities such as the average fidelity.

Alternative methods for characterizing errors include using a limited set of quantum gates in quantum hardware. For example, randomized benchmarking with Clifford gates is a significantly enhanced method for measuring the fidelity of single-qubit and two-qubit gates. However, such techniques cannot be applied to directly measure the fidelity of quantum circuits employing a universal quantum gate set. Because the quantum states produced by these limited sets of circuits may differ significantly in important respects from those produced by circuits using universal quantum gates, results obtained using such methods may therefore be considered not immediately remarkable or illustrative or non-practical. For example, Clifford circuits may not always be efficiently simulated using classical computers and may not exhibit a Porter-Thomas distribution.

As the complexity and diversity of quantum hardware increases, measuring circuit fidelity is essential for quantum circuits that are not easily and efficiently simulated by classical computers.

Systems implementing fidelity estimation for quantum computing systems as described herein can estimate the fidelity of increasingly complex quantum hardware.

The system implementing fidelity estimation for quantum computing systems as described herein is applicable to both digital models of quantum computation and analog models of quantum computation. In the case of digital models of quantum computation, the system does not require the use of a limited set of quantum gates. For example, the system may utilize complex random quantum circuits composed of gates chosen from a universal set of quantum gates, thus allowing direct measurement of quantum circuit fidelity for quantum circuits of increasing complexity with an increasing number of qubits and quantum gates. Similarly, in the case of analog models of quantum computation, the system allows for direct determination of the fidelity of quantum hardware that implements continuous Hamiltonian evolution.

Unlike other systems and methods for estimating the fidelity of quantum circuits, systems implementing fidelity estimation for quantum computing systems as described herein allow the fidelity of a quantum circuit to be estimated without the need for measurements to be made of several powers of the number of qubits.

Details of one or more implementations of the subject matter herein are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, drawings, and claims.

FIG. 1 illustrates an example system for fidelity estimation. FIG. 1 illustrates an example system for fidelity estimation. 1 is a flow diagram of an exemplary process for benchmarking the fidelity of a quantum circuit. 1 is a flow diagram of an exemplary process for fitting determined output information of a quantum system to calculated output information of a quantum system to estimate the fidelity of a quantum circuit. 1 is a flow diagram of an exemplary process for benchmarking the fidelity of quantum hardware. 1 is a flow diagram of an exemplary process for fitting determined output information of a quantum system to calculated output information of a quantum system to estimate the fidelity of the quantum hardware.

Like reference numbers and names in the various drawings indicate like elements.

This specification describes a method and system for fidelity benchmarking of quantum hardware, e.g., quantum circuits. A random instance of a quantum circuit from a given universal family of available quantum gates is selected, and statistics of the selected quantum circuit for chosen measurements are numerically calculated using classical simulation. A sequence of runs of the same quantum circuit is performed, and measurements of the quantum hardware are made. The fidelity of the quantum circuit is estimated using the numerically obtained expected statistics and the experimentally determined statistics.

1A illustrates an exemplary system 100 for fidelity estimation. The exemplary system 100 is an example of a system implemented as a classical or quantum mechanical computer program on one or more classical or quantum computing devices, at one or more locations where the systems, components, and techniques described below may be implemented.

The system includes quantum hardware 102 in communication with a fidelity estimation system 104. The quantum hardware 102 includes a quantum system that may include one or more qubits 106, e.g., qubit 108. The one or more qubits may be used to perform an algorithmic operation or a quantum computation. The particular implementation of the one or more qubits depends on the type of algorithmic operation or quantum computation that the quantum computing device is performing. For example, the qubits may include qubits implemented with atomic, molecular, or solid-state quantum systems. In other examples, the qubits may include, but are not limited to, superconducting qubits or semiconductor qubits. For clarity, four qubits are shown in FIG. 1A, but the system may include a smaller or larger number of qubits.

Each of the one or more qubits 106 may interact with one or more other qubits, for example through respective controllable couplings. In some examples, one or more qubits 106 may undergo nearest-neighbor interactions.

The one or more qubits 106 may be arranged in a variety of ways. The particular arrangement of the one or more qubits may depend on the algorithmic operation or quantum computation that the qubits are being used to perform. In some examples, the qubits may be arranged in a one-dimensional array, e.g., a chain. In other examples, the qubits may be arranged in a two-dimensional array, e.g., a lattice. For clarity, four qubits are shown in a one-dimensional array in FIG. 1, although the system may arrange the qubits in other ways.

The quantum hardware 102 includes a set of quantum gates 110. The set of quantum gates 110 includes single qubit gates, e.g., quantum gate 112, and two qubit gates, e.g., quantum gate 114. A single qubit quantum gate is a quantum gate that operates on a single qubit. Exemplary single qubit gates include, but are not limited to, a Hadamard gate, a Pauli X, Y, or Z gate, or a phase shift gate. A two qubit quantum gate is a quantum gate that operates on two qubits. Exemplary two qubit gates include, but are not limited to, a swap gate, a control gate, a Toffoli gate, or a Fredkin gate.

The set of quantum gates 110 may include a universal set of quantum gates. A universal set of quantum gates is a set of gates to which any computational operation possible on a quantum computing device can be reduced. For example, one example of a universal set of single and two-qubit quantum gates includes the Hadamard gate, the π/8 gate, and the controlled NOT gate.

The set of quantum gates 110 may be sampled to define one or more quantum circuits, e.g., quantum circuit 116. For clarity, quantum circuit 116 includes a fixed number of representative quantum gates, e.g., quantum gates 112 and 114, although a quantum circuit defined by a sampled set of quantum gates may include more or fewer quantum gates in various arrangements.

The sampled quantum circuit 116 receives as input a quantum system, e.g., one or more qubits, prepared in an initial state 122, e.g., a basis state. The quantum circuit operates on the quantum system and outputs a quantum system in a final state 120, where the final state is determined by the operations performed on the quantum system by the quantum circuit.

Each quantum gate in the set of quantum gates 110 and each quantum circuit defined by a subset of the set of quantum gates 110 is associated with a respective gate fidelity or quantum circuit fidelity. Quantum gate fidelity and quantum circuit fidelity indicate a measure of how reliably a gate or circuit converts an input into an expected output. For example, a Pauli X quantum gate operates on a single qubit and maps a 0 state to a 1 state and a 1 state to a 0 state. The fidelity of a Pauli X gate may include a number between 0 and 1 indicating how accurately and reliably the mapping is achieved, for example, whether the gate reliably maps a 0 state to a 1 state and vice versa.

The quantum hardware 102 includes one or more measurement devices 118, such as measurement device 124. The measurement devices 118 may operate on the quantum system to determine a characteristic of the quantum system, such as measurement device 124 operating on the quantum system at a final state 120.

The fidelity estimation system 104 may include a classical or quantum mechanical processing device and is in communication with the quantum hardware 102. The fidelity estimation system 104 may be configured to access a set of quantum gates 110 and sample a subset of quantum gates from the set of quantum gates 110 to define a respective quantum circuit, e.g., quantum circuit 116. The fidelity estimation system 104 may cause the defined quantum circuit to be repeatedly applied to a quantum system, e.g., one or more qubits 106, and perform respective measurements on the quantum system to determine output information, e.g., statistics, of the quantum system using one or more measurement devices 118.

The fidelity estimation system 104 may be configured to calculate output information, e.g., expected statistics, of the quantum system based on application of the quantum circuit to the quantum system, e.g., after applying a defined quantum circuit on the quantum system, and to fit the determined output information of the quantum system to the calculated output information of the quantum system to estimate the fidelity of the quantum circuit. Estimating the fidelity of the quantum circuit is described in more detail below with respect to Figures 2 and 3.

FIG. 1B illustrates an exemplary system 150 for fidelity estimation. The exemplary system 150 is an example of a system implemented as a classical or quantum mechanical computer program on one or more classical or quantum computing devices at one or more locations where the systems, components, and techniques described below may be implemented.

The system includes quantum hardware 152 in communication with a fidelity estimation system 154. The quantum hardware 152 includes a quantum system that may include one or more qubits 156, e.g., qubit 158. As described above with respect to FIG. 1A, the one or more qubits may be used to perform an algorithmic operation or a quantum computation. The particular implementation of the one or more qubits depends on the type of algorithmic operation or quantum computation that the quantum computing device is performing. For example, the qubits may include qubits implemented with atomic, molecular, or solid-state quantum systems. In other examples, the qubits may include, but are not limited to, superconducting qubits or semiconductor qubits. For clarity, four qubits are shown in FIG. 1B, but the system may include a smaller or larger number of qubits.

Each of the one or more qubits 156 may interact with one or more other qubits, for example through respective controllable couplings. In some examples, one or more qubits 156 may undergo nearest-neighbor interactions.

The one or more qubits 156 may be arranged in a variety of ways. The particular arrangement of the one or more qubits may depend on the algorithmic operation or quantum computation that the qubits are being used to perform. In some examples, the qubits may be arranged in a one-dimensional array, e.g., a chain. In other examples, the qubits may be arranged in a two-dimensional array, e.g., a lattice. For clarity, four qubits are shown in a one-dimensional array in FIG. 1, although the system may arrange the qubits in other ways.

The quantum hardware 152 includes one or more components 160 for performing continuous Hamiltonian evolution. The one or more components 160 for performing continuous Hamiltonian evolution may implement one or more Hamiltonians, which in turn may determine a unitary operator that defines the evolution of the quantum system to which it is applied. For example, the component 160 may implement a Hamiltonian H, which in turn gives rise to a unitary operator 162. The unitary operator 162 defines the evolution of a quantum system that is provided at an initial state 164 and becomes a final state 166 of the quantum system.

Each successive Hamiltonian evolution of a quantum system is associated with a respective quantum hardware fidelity, which is a measure of how reliably the hardware transforms inputs into expected outputs, as explained above with respect to Figure 1A.

Quantum hardware 152 includes one or more measurement devices 168, such as measurement device 172. Measurement device 168 may operate on the quantum system to determine a property of the quantum system, such as measurement device 172 operating on the quantum system at final state 166.

The fidelity estimation system 154 may include a classical or quantum mechanical processing device and is in communication with the quantum hardware 152. The fidelity estimation system 154 may be configured to access the quantum hardware 512 and select a particular component 160 for the continuous Hamiltonian evolution. The fidelity estimation system 154 may cause the quantum hardware to repeatedly perform the continuous Hamiltonian evolution corresponding to the selected component on the quantum system, e.g., one or more qubits 156, and perform respective measurements on the quantum system to determine output information of the quantum system.

The fidelity estimation system 154 may be configured to calculate output information of the quantum system based on performing a selected continuous Hamiltonian evolution on the quantum system, and to fit the determined output information of the quantum system to the calculated output information of the quantum system to estimate the fidelity of the quantum hardware. Estimating the fidelity of the quantum hardware is described in more detail below with respect to Figures 4 and 5.

Programming the Hardware Figure 2 is a flow chart of an exemplary process 200 for benchmarking the fidelity of a quantum circuit. For convenience, process 200 is described as being performed by a system of one or more classical or quantum mechanical computing devices at one or more locations. For example, a quantum computing system, such as quantum computing system 100 of Figure 1A, suitably programmed in accordance with the present specification, can perform process 200.

The system accesses a set of quantum gates (step 202). The set of quantum gates may include one or more single-qubit gates. Each quantum gate in the set of quantum gates may be associated with a respective quantum gate fidelity. The set of quantum gates may include one or more two-qubit gates. Exemplary single and two-qubit gates are described above with respect to FIG. 1A. In some implementations, the set of quantum gates includes a universal set of quantum gates. The universal set of quantum gates is described above with respect to FIG. 1A.

The system samples a subset of quantum gates from the set of quantum gates (step 204). In some implementations, the system may randomly sample a subset of quantum gates from the set of quantum gates. The sampled subset of quantum gates defines a quantum circuit. For example, by randomly sampling a subset of quantum gates from the available set of quantum gates, the system generates a random instance of a quantum circuit.

In some implementations, the system may repeatedly sample a set or subset of quantum gates, with each sampled subset defining a respective quantum circuit. The system may sample subsets of quantum gates that include a same number of quantum gates of comparable respective quantum gate fidelity, e.g., instances of a quantum circuit defined by the sampled subsets of quantum gates may include a same number of quantum gates of comparable quantum gate fidelity. By sampling subsets of quantum gates that include a same number of quantum gates of comparable respective quantum gate fidelity, the system may improve the consistency of results obtained by process 200 and avoid large systematic errors.

The system applies the quantum circuit to the quantum system and performs measurements on the quantum system to determine output information, e.g., statistics of the quantum system (step 206). For example, the system may include or possibly access a quantum system, e.g., a quantum system including one or more qubits as shown in FIG. 1A, and repeatedly apply a quantum circuit defined by a sampled subset of quantum gates to an initialized state of the quantum system. For each application of the quantum circuit, the system may perform a respective measurement on the quantum system and use the measurement results to determine output information of the quantum system. As an example, the system may sample a subset of quantum gates defining each quantum circuit, perform m runs of the quantum circuit on the quantum system of interest, and obtain a sequence of computationally measured bit strings {x ₁ , x ₂ , ..., x _m }.

As described above, in some implementations, the system may repeatedly sample subsets of quantum gates from the set of quantum gates, with each sampled subset defining a respective quantum circuit. In such cases, the system may repeatedly apply each sampled quantum circuit to the quantum system, and perform respective measurements on the quantum system to determine respective statistics of the quantum system for each of the sampled circuits.

The system calculates output information, e.g., expected, e.g., exact or ideal statistics, of the quantum system based on applying the quantum circuit to the quantum system (step 208). For example, the system may use available computational techniques, e.g., classical computing techniques, to calculate the output information of the quantum system after applying the quantum circuit to the quantum system.

Continuing with the above example, to determine the output information of the quantum system, the system may compute a set of probabilities {p(y _j )} corresponding to the probability of obtaining each possible bit string y _j . As described above with respect to step 204, in some implementations, the system may repeatedly sample subsets of quantum gates from the set of quantum gates, with each sampled subset defining a respective quantum circuit. In such a case, the system may compute, for each sampled quantum circuit, a set of probabilities corresponding to the probability of obtaining each possible bit string.

The system estimates the fidelity of the quantum circuit based on the determined output information and the calculated output information of the quantum system (step 210). The system may estimate the fidelity of the quantum circuit by fitting the determined output information of the quantum system to the calculated output information of the quantum system. The system may fit the results of the measurements made in step 206 to a statistical mixture of (i) the output information of the quantum system based on application of the quantum circuit to the quantum system as calculated in step 208, and (ii) the fully mixed quantum state. Fitting the determined output information of the quantum system to the calculated output information of the quantum system to estimate the fidelity of the quantum circuit is described in more detail below with respect to FIG. 3.

If the system repeatedly samples a subset of quantum gates to define multiple quantum circuits, the system fits the respective determined output information for the quantum systems to the respective calculated output information of the quantum systems to estimate the respective fidelity of each quantum circuit. The system may repeatedly sample the subset of quantum gates from the set of quantum gates until completion of the event, e.g., when the uncertainty of the estimated fidelity falls below a predefined threshold. By increasing the number of iterations, the uncertainty of the estimated fidelity may be reduced to a desired threshold of certainty, e.g., according to the inverse square root law of the number of iterations.

FIG. 3 is a flowchart of an exemplary process 300 for fitting determined output information, e.g., statistics, of a quantum system to calculated output information, e.g., statistics, of a quantum system to estimate the fidelity of a quantum circuit. For convenience, process 300 is described as being performed by a system of one or more classical or quantum mechanical computing devices at one or more locations. For example, one or more classical processors suitably programmed in accordance with this specification, e.g., classical processor 104 of FIG. 1A, may perform process 300.

The system defines (step 302) a convex combination of the calculated output information of the quantum system and the fully mixed quantum states described above with respect to step 208 of Figure 2. For example, the convex combination may be given by the following equation (1):
In equation (1), α represents the fidelity of the quantum circuit, |ψ〉 represents the calculated quantum state of the quantum system based on the application of the quantum circuit to the quantum system, and I/N represents the completely mixed state with dimension N of the Hilbert space containing |ψ〉.

The system estimates the fidelity of the quantum circuit by comparing the defined convex combination in equation (1) to the determined output information of the quantum system as described above with respect to step 206 of FIG. 2 (step 304). For example, the system may estimate the fidelity of the quantum circuit by comparing the defined convex combination to the determined output information of the quantum system and determining the value of the parameter α.

Continuing with the example given above in FIG. 2, the system may generate a quantity, for each sampled subset of quantum gates, for a corresponding experimentally obtained sequence of bit strings {x ₁ ,x ₂ ,...,x _m }.
may be calculated numerically. Under certain assumptions, e.g., the quantum circuit is long enough (the circuit depth does not grow faster than n ^1/D to possible corrections in log(n), where n is the number of qubits and D is the dimension of the qubit array, e.g., as shown in FIG. 1A for a 1D array of qubits, D=1, depth=n, or for a constant depth or logarithmic depth of n for a hypothetical configuration in which a two-qubit gate is performed between any pair of qubits, D may be infinite), the parameter α, e.g., the quantum circuit fidelity, may be estimated by α=c+ln(N)+γ, where γ is Euler's constant and c is defined above. In some implementations, the error in estimating α may be given by k/m ^1/2 , where k≈1. This may represent the number of measurements required, and is independent of the number of qubits.

In some implementations, the system may estimate the fidelity of a quantum circuit through a numerical comparison to any number of statistical aggregates. For example,
Instead of using
One may use, for example, the sum of squared probabilities. Other quantities may also be used. The requirement is that the quantity is a statistical set that may be computed using simulation of the quantum circuit, and that the quantity must be equally sensitive to errors in the physical implementation of the quantum circuit.

As described above, in some implementations, the system may repeatedly sample a subset of quantum gates and repeatedly estimate the corresponding fidelity for a quantum circuit defined by the subset of quantum gates. By repeatedly estimating the fidelity for a quantum circuit, the system can increase the reliability of the fidelity estimates and reduce the likelihood that systematic errors or correlations will affect the resulting results. As a simple example, by repeatedly estimating the fidelity, the system can determine that the quantum hardware is functioning as intended.

FIG. 4 is a flow chart of an exemplary process 400 for benchmarking the fidelity of quantum hardware. For convenience, process 400 is described as being performed by a system of one or more classical or quantum computing devices at one or more locations. For example, a quantum computing system, such as quantum computing system 150 of FIG. 1B, suitably programmed in accordance with the present specification, may perform process 400.

The system accesses quantum hardware (step 402). The quantum hardware may be configured to perform one or more different successive Hamiltonian evolutions, as described above with respect to FIG. 1B. Each successive Hamiltonian evolution may be associated with a respective quantum hardware fidelity.

The system selects a particular continuous Hamiltonian evolution (step 404). For example, the system may select one or more components that are configured to implement a particular Hamiltonian that determines the unitary operator that defines the evolution of the quantum system to which it is applied. In some cases, the system may randomly select a particular continuous Hamiltonian evolution.

In some implementations, the system may iteratively select a continuous Hamiltonian evolution. For example, in some cases, the quantum hardware may be configured to perform one or more different continuous Hamiltonian evolutions on one or more interacting qubits, where each qubit interaction has a respective fidelity associated with it. In these cases, the system may iteratively select a continuous Hamiltonian evolution that includes interactions of comparable fidelity. By selecting a continuous Hamiltonian evolution that includes qubit interactions of comparable fidelity, the system can improve the consistency of results obtained by process 400 and avoid large systematic errors.

The system performs a selected successive Hamiltonian evolution of the quantum system and performs measurements on the quantum system to determine an output information value of the quantum system (step 406). For example, the system may include or possibly access a quantum system, e.g., a quantum system including one or more qubits as shown in FIG. 1B, and iteratively perform the selected successive Hamiltonian evolution by selecting one or more corresponding components that implement a Hamiltonian that gives rise to a unitary operator that defines the evolution of the quantum system, as described above with respect to FIG. 1B. For each evolution of the quantum system, the system may perform a respective measurement on the quantum system and use the measurement results to determine the output information of the quantum system.

As described above, in some implementations, the system may iteratively select successive Hamiltonian evolutions. In such cases, the system may iteratively perform each selected successive Hamiltonian evolution of the quantum system and perform respective measurements on the quantum system to determine respective output information of the quantum system for the selected successive Hamiltonian evolution.

The system calculates the output information of the quantum system on the quantum system based on performing the selected continuous Hamiltonian evolution (step 408). For example, the system may use available computational techniques, e.g., classical computing techniques, to calculate the output information of the quantum system after the selected continuous evolution of the quantum system.

The system estimates the fidelity of the quantum hardware based on the determined output information and the calculated output information of the quantum system (step 410). The system may estimate the fidelity of the quantum hardware by fitting the determined output information of the quantum system to the calculated output information of the quantum system. The system may fit the results of the measurements made in step 406 to a statistical mixture of (i) the output information of the quantum system based on performing the selected continuous Hamiltonian evolution on the quantum system as calculated in step 408, and (ii) the fully mixed quantum state. Fitting the determined output information of the quantum system to the calculated output information of the quantum system to estimate the fidelity of the quantum hardware is described in more detail below with respect to FIG. 5.

If the system repeatedly selects the continuous Hamiltonian evolution, the system fits the respective determined output information for the quantum system to the respective calculated output information of the quantum system to estimate the respective fidelity of each quantum hardware corresponding to each selected continuous Hamiltonian evolution. The system may repeatedly select the continuous Hamiltonian evolution until the completion of the event, for example, when the uncertainty of the estimated fidelity falls below a predetermined threshold. By increasing the number of iterations, the uncertainty of the estimated fidelity may be reduced to a desired threshold of certainty, for example according to the inverse square root law of the number of iterations.

FIG. 5 is a flowchart of an exemplary process 500 for fitting determined output information of a quantum system to calculated output information of a quantum system to estimate the fidelity of the quantum hardware. For convenience, process 500 is described as being performed by a system of one or more classical or quantum mechanical computing devices at one or more locations. For example, one or more classical processors suitably programmed in accordance with this specification, such as classical processor 104 of FIG. 1, may perform process 500.

The system defines (step 502) a convex combination of the calculated output information of the quantum system and the fully mixed quantum states described above with respect to step 408 of Figure 4. For example, the convex combination may be given by the following equation (1):
In equation (1), α represents the fidelity of the quantum circuit, |ψ〉 represents the calculated quantum state of the quantum system based on performing continuous Hamiltonian evolution on the quantum system, and I/N represents the completely mixed state with dimension N of the Hilbert space containing |ψ〉.

The system estimates the fidelity of the quantum hardware by comparing the defined convex combination in equation (1) with the determined output information of the quantum hardware as described above with respect to step 406 of FIG. 4 (step 504). For example, the system may estimate the fidelity of the quantum hardware by comparing the defined convex combination with the determined output information of the quantum system and determining the value of the parameter α.

As described above, in some implementations, the system may iteratively select a continuous Hamiltonian evolution and iteratively estimate the corresponding fidelity for the quantum hardware corresponding to the continuous Hamiltonian evolution. By iteratively estimating the fidelity for the quantum hardware, the system can increase the reliability of the fidelity estimate and reduce the likelihood that systematic errors or correlations affect the results obtained. As a simple example, by iteratively estimating the fidelity, the system can determine that the quantum hardware is functioning as intended.

The implementations of the digital and/or quantum objects and digital function operations and quantum operations described herein may be implemented in digital electronic circuitry, suitable quantum circuitry, or more generally, in a quantum computing system, in tangibly embodied digital and/or quantum computer software or firmware, in digital and/or quantum computer hardware including the structures disclosed herein and their structural equivalents, or in one or more combinations thereof. 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.

Implementations of the digital and/or quantum objects described herein may be implemented as one or more digital and/or quantum computer programs, e.g., as one or more modules of digital and/or quantum computer program instructions encoded on a tangible, non-transitory storage medium for execution by or to control the operation of a data processing device. The digital and/or quantum computer storage medium may 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 of one or more of the above. Alternatively or additionally, the program instructions may be encoded on an artificially generated propagated signal capable of encoding digital and/or quantum information, e.g., a machine-generated electrical, optical, or electromagnetic signal generated to encode digital and/or quantum information for transmission to a receiver device suitable for execution by a data processing device.

The terms quantum information and quantum data refer to information or data carried by and held or stored in a quantum system, where the smallest non-trivial system is a qubit, e.g., a system that defines a unit of quantum information. The term "qubit" is understood to encompass all quantum systems that may be appropriately approximated as two-level systems in the corresponding context. Such quantum systems may include multi-level systems, e.g., having two or more levels. By way of example, such systems may include atomic qubits, electronic qubits, photon qubits, ion qubits, or superconducting qubits. In many implementations, the computational basis state is identified with the ground, first excited state, although it is understood that other setups are conceivable in which the computational state is identified with a higher level excited state. The term "data processing apparatus" refers to digital and/or quantum data processing hardware and encompasses all kinds of apparatus, devices, machines for processing digital and/or quantum data, including, by way of example, programmable digital processors, programmable quantum processors, digital computers, quantum computers, multiple digital and quantum processors or computers, and combinations thereof. The device may be or may further include special purpose logic circuitry, e.g., an FPGA (field programmable gate array), an ASIC (application specific integrated circuit), or a quantum simulator, e.g., a quantum data processing device designed to simulate or generate information about a particular quantum system. In particular, a quantum simulator is a special purpose quantum computer that is not capable of universal quantum computing. In addition to hardware, the device may optionally include code that creates an execution environment for digital and/or quantum computer programs, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of these.

A digital computer program, which may be referred to or described as a program, software, software application, module, software module, script, or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and the digital computer program may be deployed in any form, including as a stand-alone program, or as a module, component, subroutine, or other unit suitable for use in a computing environment. A quantum computer program, which may be referred to or described as a program, software, software application, module, software module, script, or code, may be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and may be converted into a suitable quantum programming language, or may be written in a quantum programming language, e.g., QCL or Quipper.

A digital and/or quantum computer program may, but need not, correspond to a file in a file system. A program may be stored in a portion of a file with other programs or data, e.g., one or more scripts stored in a markup language document, in a single file dedicated to the program, or in multiple coordinated files, e.g., files storing one or more modules, subprograms, or portions of code. A digital and/or quantum computer program may be deployed to run on one digital or quantum computer, or on multiple digital and/or quantum computers located at one site or distributed across multiple sites and interconnected by a digital and/or quantum data communication network. A quantum data communication network is understood to be a network that may transmit quantum data using quantum systems, e.g., qubits. Generally, a digital data communication network cannot transmit quantum data, but a quantum data communication network may transmit both quantum data and digital data.

The processes and logic flows described herein may be performed by one or more programmable digital and/or quantum computers, suitably operating on one or more digital and/or quantum processors, executing one or more digital and/or quantum computer programs to perform functions by operating on input digital and quantum data and generating outputs. The processes and logic flows may also be performed by, and an apparatus may be implemented as, special purpose logic circuitry, e.g., an FPGA or ASIC, or a quantum simulator, or a combination of special purpose logic circuitry or a quantum simulator and one or more programmed digital and/or quantum computers.

When one or more digital and/or quantum computer systems are "configured to" perform a particular operation or action, it means that the system has installed thereon software, firmware, hardware, or a combination thereof that, when operated, causes the system to perform the operation or action. When one or more digital and/or quantum computers are configured to perform a particular operation or action, it means that one or more programs contain instructions that, when executed by the digital and/or quantum data processing device, cause the device to perform the operation or action. A quantum computer may also receive instructions from a digital computer that, when executed by the quantum computing device, causes the device to perform an operation or action.

A digital and/or quantum computer suitable for executing a digital and/or quantum computer program can be based on a general purpose or dedicated digital and/or quantum processor, or both, or any other kind of central digital and/or quantum processing unit. Typically, the central digital and/or quantum processing unit will receive instructions and digital and/or quantum data from a read-only memory, a random access memory, or a quantum system suitable for transmitting quantum data, e.g. photons, or a combination thereof.

The essential elements of a digital and/or quantum computer are a central processing unit for performing or executing instructions and one or more memory devices for storing instructions and digital and/or quantum data. The central processing unit and memory may be supplemented by or incorporated into dedicated logic circuits or quantum simulators. Typically, a digital and/or quantum computer will also include one or more mass storage devices for storing digital and/or quantum data, such as magnetic, magneto-optical, optical disks, or quantum systems suitable for storing quantum information, or will be operatively coupled to receive digital and/or quantum data from these mass storage devices, or to transfer digital and/or quantum data to these mass storage devices, or both. However, a digital and/or quantum computer need not have such devices.

Digital and/or computer readable media suitable for storing digital and/or quantum computer program instructions and digital and/or quantum data include, by way of example, all forms of non-volatile digital and/or quantum memories, media, memory devices, including semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices, magnetic disks, e.g., internal hard disks or removable disks, magneto-optical disks, CD-ROM and DVD-ROM disks, and quantum systems, e.g., trapped atoms or electrons. Quantum memories are understood to be devices capable of storing quantum data with high fidelity and efficiency for long periods of time, e.g., light-matter interfaces where light is used to transmit and materialize quantum features of quantum data, such as superposition or quantum coherence.

The control of the various systems described herein, or portions thereof, may be implemented in a digital and/or quantum computer program product that includes 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 described herein, or portions thereof, may each be implemented as an apparatus, method, or system, which may include one or more digital and/or quantum processing devices and a memory for storing instructions executable to perform the operations described herein.

Although the specification includes details of many specific implementations, these should not be construed as limitations on the scope of what may be claimed, but rather as descriptions of features that may be specific to a particular implementation. Some features described in the context of separate implementations herein may also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation may also be implemented in multiple implementations separately or in any suitable subcombination. Furthermore, although features may be described above as functioning in a combination and may even initially be claimed as such, one or more features from a claimed combination may in some cases be deleted from the combination, and the claimed combination may be directed to a subcombination or a variation of a subcombination.

Similarly, although operations are shown in the figures in a particular order, this should not be understood as requiring such operations to be performed in the particular order or sequence shown, or that all of the illustrated operations be performed, to achieve desirable results. In some environments, multitasking and parallel processing may be advantageous. Furthermore, the separation of various system modules and components in the implementations described above should not be understood as requiring such separation in all implementations, and it should be understood that the program components and systems described may generally be integrated together in a single software product or packaged in multiple software products.

Specific implementations of the subject matter have been described. Other implementations are within the scope of the claims. For example, the actions recited in the claims may be performed in a different order and still achieve desirable results. As an example, the processes depicted in the accompanying figures do not necessarily require the particular order depicted, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

100 Systems
102 Quantum Hardware
104 Fidelity Estimation System
106 Cubits
108 Cubits
110 Quantum Gate
112 Quantum Gates
114 Quantum Gates
116 Quantum Circuits
118 Measuring Devices
120 Final state
124 Measuring Devices
150 Systems
152 Quantum Hardware
154 Fidelity Estimation System
156 Cubits
158 Cubits
160 Components
162 Unitary Operators
164 Initial state
166 Final State
168 Measuring Devices
172 Measuring Devices

Claims (9)

1. A computer-implemented method for fidelity benchmarking of quantum hardware comprising a quantum system, comprising:
receiving measurement data from the quantum hardware to determine output information , the measurement data being obtained through measurement of the quantum system after a random quantum circuit consisting of gates from a universal set of quantum gates is applied to the quantum system;
simulating application of the random quantum circuit to the quantum system to determine expected statistics of the quantum system;
and estimating a fidelity of the random quantum circuit based on the determined output information and the determined expected statistics of the quantum system. The method of claim 1, wherein estimating the fidelity of the random quantum circuit based on the determined output information and the determined expected statistics of the quantum system comprises fitting the determined output information of the quantum system to a convex combination of i) the determined expected statistics of the quantum system and ii) a fully mixed quantum state. The method of claim 1, wherein estimating the fidelity of the random quantum circuit based on the determined output information and the expected statistics of the quantum system includes comparing a convex combination of the determined expected statistics and a fully mixed quantum state of the quantum system to the determined output information of the quantum system. The method of claim 3, wherein comparing the convex combination of the determined expected statistics and fully mixed quantum states of the quantum system to the determined output information of the quantum system comprises fitting the determined output information of the quantum system to the convex combination of the determined expected statistics and fully mixed quantum states of the quantum system. The convex combination is
4. The method of claim 2 or 3, wherein α represents the fidelity of a quantum circuit, |ψ〉 represents a simulated quantum state of the quantum system based on application of the quantum circuit to the quantum system, and I/N represents the fully mixed quantum state. The method of claim 5, wherein fitting the determined output information of the quantum system to the determined expected statistics and the convex combination of the fully mixed quantum state of the quantum system comprises solving for α. The method of claim 1, wherein the set of quantum gates includes a single-qubit quantum gate and a two-qubit quantum gate. A system comprising one or more computers and one or more storage devices storing instructions, the instructions being operable, when executed by the one or more computers, to cause the one or more computers to perform operations according to the method of any one of claims 1 to 7. One or more computer-readable storage media storing instructions executable by a processing device that, when executed, cause the processing device to perform operations in accordance with the method of any one of claims 1 to 7.