Efficient quantum memories via shadow tomography

Abstract

One example aspect of the present disclosure is directed to directed to a method for measuring a first quantum state via a quantum computing system (QCS). The method includes receiving a first copy of the first quantum state. A first copy of a second quantum state is received. The second quantum state is a conjugate state of the first quantum state. A first value is determined. Determining the first value may is on measuring, via the QCS, a first observable of the first copy of the first quantum state. A second value is determined. Determining the second value is based on measuring, via the QCS, the first observable of the first copy of the second quantum state. An approximation of the first quantum state is determined. Determining the approximation of the first quantum state is based on the first value and the second value.

Description

EFFICIENT QUANTUM MEMORIES VIA SHADOW TOMOGRAPHYPRIORITY

[0002] This application claims priority to U.S. Provisional Application No 63 / 616,462, entitled EFFICIENT QUANTUM MEMORIES VIA SHADOW TOMOGRAPHY, filed on December 29, 2023, the contents of which of herein incorporated in their entirety.FIELD

[0003] The present disclosure relates generally to quantum computing and information processing systems, and more particularly to efficient quantum memories via shadow tomography.BACKGROUND

[0004] Quantum computing is a computing method that takes advantage of quantum effects, such as superposition of basis states and entanglement to perform certain computations more efficiently than a classical digital computer. In contrast to a digital computer, which stores and manipulates information in the form of bits, e.g., a “1” or “0,” quantum computing systems can manipulate information using quantum bits (“qubits”). A qubit can refer to a quantum device that enables the superposition of multiple states, e.g., data in both the “0” and “1” state, and / or to the superposition of data, itself, in the multiple states. In accordance with conventional terminology, the superposition of a “0” and “1” state in a quantum system may be represented, e.g., as a |0) + b |1) The “0” and “1” states of a digital computer are analogous to the |0) and |1) basis states, respectively of a qubit.SUMMARY

[0005] Aspects and advantages of embodiments of the present disclosure will be set forth in part in the following description, or can be learned from the description, or can be learned through practice of the embodiments.

[0006] One example aspect of the present disclosure is directed to a method for measuring a first quantum state via a quantum computing system (QCS). The method includes receiving a first copy of the first quantum state. A first copy of a second quantum state is received. The second quantum state is a conjugate state of the first quantum state. A first value is determined. Determining the first value may is on measuring, via the QCS, a first observable of the first copy of the first quantum state. A second value is determined. Determining the second value is based onmeasuring, via the QCS, the first observable of the first copy of the second quantum state. An approximation of the first quantum state is determined. Determining the approximation of the first quantum state is based on the first value and the second value.

[0007] Other aspects of the present disclosure are directed to various systems, methods, apparatuses, non-transitory computer-readable media, computer-readable instructions, and computing devices.

[0008] These and other features, aspects, and advantages of various embodiments of the present disclosure will become better understood with reference to the following description and appended claims. The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate example embodiments of the present disclosure and, together with the description, explain the related principles.BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Detailed discussion of embodiments directed to one of ordinary skill in the art is set forth in the specification, which refers to the appended figures, in which:

[0010] FIG. 1 depicts an example quantum computing system according to example embodiments of the present disclosure.

[0011] FIG. 2 shows an algorithm for learning all displacement amplitudes up to a possible minus sign, according to various embodiments.

[0012] FIG. 3A shows an algorithm for determining the signs of displacement amplitudes, according to various embodiments.

[0013] FIG. 3B shows an algorithm for finding a hypothesis state, according to various embodiments.

[0014] FIG. 4 shows a method for determining an approximation of a quantum state, according to various embodiments.DETAILED DESCRIPTION

[0015] Example aspects of the present disclosure are directed to employing shadow tomography methods to determine at least an approximation of a quantum state (e.g., a quantum state encoded in a set of qubits or another system that encodes the quantum state). The quantum state may have been generated via quantum computations within a quantum computing system (QCS), a quantum sensor (or an array of quantum sensors), or any other system that is enabled togenerate coherent quantum states encoded in a set of particles, qubits, and / or circuits (e.g., superconducting circuits that implement a qubit). Such quantum sensors may be employed in applications such as but not limited to gravity imaging, timekeeping, and / or medical detection. More specifically, an encoding of an unknown quantum state (p) and its conjugate quantum state (p*) may be received at a QCS and / or generated at the QCS. In some embodiments, one or more copies of an encoding of the quantum state and its conjugate state may be received (or generated) by a QCS or other system (e.g., a quantum sensor). The unknown quantum state may be determined via encodings of the quantum state, encodings of the conjugate state, and a shadow tomography algorithm employed by the QCS. Compared to traditional quantum tomography techniques (e.g., non-shadow tomography), the embodiments determine an approximation of the quantum state, while requiring significantly less (e.g., exponentially less) copies of an encoding of the quantum state. The embodiments achieve this significant reduction in the quantity of quantum encodings via the employment of an encoding of the quantum state, an encoding of the conjugate state, and shadow tomography techniques. Accordingly, relatively small quantum memories may be sufficient to store a copy of the quantum state and a copy of the conjugate state. Thus, the embodiments may be employed for quantum memory applications. Because significantly less copies of the encoding of the quantum state are required, the quantum memories enabled by the embodiments are significantly more efficient than traditional quantum memories.

[0016] The ability of quantum computers to directly manipulate and analyze quantum states stored in a quantum memory allows them to learn aspects of the physical universe that would otherwise be inaccessible given a modest number of measurements. Applications of shadow tomography algorithms highlight the small number of copies of a quantum state required to determine an approximation of the quantum state. In the embodiments herein, shadow tomography algorithms are employed for the setting of quantum systems which approximate bosonic cavity modes. A traditional quantum memory that stores a '-qubit quantum state (e.g., designated herein as p® , with a constant K) has exponential limitations for certain tasks in this setting. Meanwhile, a minimal (or at least reduced) quantum memory storing p ® p* e.g., where the K superscript is implied), where p* is the complex conjugate state, suffices for a computationally efficient shadow tomography scheme. Access to p* represents a novel learning resource, which can be exploited alongside the resource of quantum memory and entangled measurements. Physically, access to p ® p* allows an exponentially improved signal to noise ratio for some highly mixed optical and sensor states. Complementary to these results, the embodiments employ a version of classical shadows tailored to the setting of quantum systems designed to approximate bosonic modes. Theembodiments may be employed to develop the power of quantum computers with minimal (or at least decreased) quantum memories in uncovering new physical phenomena.

[0017] Quantum computing, quantum information, and quantum sensing technologies provide some of the most intricate control available over the physical universe. In addition to quantum computation, quantum sensors and quantum networks offer opportunities to collect and transmit “quantum pictures” (e.g., encodings of quantum states) of portions of the physical universe. It is noted that measuring a quantum state that is in superposition with respect to a particular observable necessarily destroys the quantum state, as well any information relating to other observables that do not commute with the particle observable. Thus, when the quantum pictures are generated via a quantum sensor, rather than first measuring observables of the states and generating a limited classical dataset, the embodiments may provide the quantum pictures to a QCS (e.g., via the transmission along a quantum network). In other embodiments, the quantum picture may be generated in the QCS (e.g., when simulating a quantum system via quantum analog computing). The QCS may operate on the quantum pictures directly (e.g., via quantum logic gates and / or qubit couplers). With the ability to collect and store at least two noisy copies of a quantum state (e.g., the quantum state and the conjugate of the quantum state) on a quantum device, the embodiments enable one to learn features of a state with exponentially fewer copies than would otherwise be possible. Via the embodiments, if the supply of such states is limited, even noisy devices can reveal otherwise invisible aspects of the physical universe.

[0018] As noted above, the embodiments employ shadow tomography to determine an approximation for a quantum state. More specifically the embodiments are enabled to learn the expected value of a large number of non-commuting observables (or measurements) requiring only a small number of copies of encodings of the quantum state. Learning the expected value of noncommuting observables enables inferring the approximation of the quantum state. It is shown that only a polylogarithmic number of copies of the state are required as a function of the dimension and number of measurements, even if those measurements (or observables) don’t commute.

[0019] A requirement of employing shadow tomography schemes is the usage of a quantum memory that can simultaneously store the quantum state and the conjugate state. For example, if one is forced to measure single copies of the quantum state at a time (e g., a single copy without the conjugate state), shadow tomography often becomes impossible. In addition, even when a quantum memory is present, it is not always clear how many copies of a quantum state (e.g., multiple copies of the state without the conjugate state) must be kept in memory at once in order to reap these benefits. For practical and theoretical reasons, one might wonder what ispossible given a quantum memory that is only a constant times the space required for a single copy of the state, or a minimal quantum memory.

[0020] The embodiments exploit the power of minimal or nearly minimal quantum memories. For example, in contrast to traditional tomography methods, some embodiments determine the magnitudes of qubit Pauli operators in a computationally-efficient manner with minimal (or at least decreased) quantum memory. In tomography methods, this task may require exponentially many measurements without quantum memory. In some embodiments, the determination of the signs of the Pauli operators is performed and is computationally efficient. In addition to Pauli operators, it is shown herein that a number of state distinguishing tasks are possible with nearly minimal quantum memories. The embodiments relate these particular measurements to those made by commonly developed modem quantum sensors. In contrast, traditional tomography methods tend to focus on ensembles of single qubits, bosonic cavities, or stretched single qubits attempting to achieve Heisenberg limited scaling as in GHZ states.

[0021] The embodiments expand the classes of operators for which a fully efficient scheme is known and for which a modified minimal quantum memory is sufficient. Throughout, operators native to a d-dimensional quantum system, which approximates a continuous bosonic mode, are discussed. This setting is closely related to current quantum sensor designs. Here, the natural basis of operators are the displacement operators, or Heisenberg-Weyl operators, which generalize the Pauli operators to d-dimensions. The embodiments employ a shadow tomography scheme for this class of operators. Some embodiments employ a learning algorithm that requires access to the complex conjugate state p*. It is shown that for this class of operators, a minimal (or at least decreased) quantum memory containing the quantum state and its conjugate state (e.g., p ® p‘] is exponentially more powerful than the standard quantum memory encoding only p®Kfor a quantum state p, even when K grows with the desired precision. While the minimal quantum memory is sufficient to learn the expected values of these operators up to a sign, learning the sign remains may require additional computation. As described below, the quantum memories of the embodiments that store quantum states and their conjugate states is sufficient in an information theoretic sense to learn the amplitudes and signs of various displacement observables. In addition to learning displacement operators using p ® p*, some embodiments employ a version of classical shadows tailored to the setting of a d-dimensional quantum system, where no quantum memory is necessary.

[0022] In some embodiments, a first copy of a first quantum state is received. A first copy of a second quantum state may also be received. The second quantum state may be a conjugatestate of the first quantum state. A first value may be determined. Determining the first value may be based on the QWCS measuring a first observable of the first copy of the first quantum state. A second value may be determined. Determining the second value may be determined based on the QCS measuring the first observable of the first copy of the second quantum state. An approximation of the first quantum state may be determined based on the first value and the second value.

[0023] In some embodiments, a first set of quantum states may be received. Each quantum state of the first set of quantum states is a copy of the first quantum state. The first set of quantum states includes the first copy of the first quantum state. A second set of quantum states is received. Each quantum state of the second set of quantum states is a copy of the second quantum state, which is the conjugate state of the first quantum state. The second set of quantum states includes first copy of the second quantum state. A first set of values is determined. Each value of the first set of values corresponds to a separate quantum state of the first set of quantum states.Determining each value of the first set of values may be based on the QCS measuring the first observable of the quantum state of the first set of quantum states that corresponds to the value of the first set of values. The first set of values includes the first value. A second set of values is determined. Each value of the second set of values corresponds to a separate quantum state of the second set of quantum state. Determining each value of the second set of values is based on the QCS measuring the first observable of the quantum state of the second set of quantum states that corresponds to the value of the second set of values. The second set of values includes the second value. In these embodiments, the approximation of the first quantum state is further based on the first set of values and the second set of values.

[0024] Aspects of the present disclosure provide a number of technical effects and benefits. For instance, the embodiments are enabled to determine an approximation of a quantum state with significantly less copies of the quantum state than traditional methods. As such, the embodiments may be applied to develop quantum memories that are significantly more efficient (e g., smaller) than traditional quantum memories.Quantum Computing Systems

[0025] FIG. 1 depicts an example quantum computing system 100. The system 100 is an example of a system of one or more classical computers and / or quantum computing devices in one or more locations, in which the systems, components, and techniques described below can be implemented. Those of ordinary skill in the art, using the disclosures provided herein, willunderstand that other quantum computing devices or systems can be used without deviating from the scope of the present disclosure.

[0026] The system 100 includes quantum hardware 102 in data communication with one or more classical processors 104. The classical processors 104 can be configured to execute computer-readable instructions stored in one or more memory devices to perform operations, such as any of the operations described herein. The quantum hardware 102 includes components for performing quantum computation. For example, the quantum hardware 102 includes a quantum system 110, control device(s) 112, and readout device(s) 114 (e.g., readout resonator(s)). The quantum system 110 can include one or more multi-level quantum subsystems, such as a register of qubits (e.g., qubits 120). In some implementations, the multi-level quantum subsystems can include superconducting qubits, such as flux qubits, charge qubits, transmon qubits, gmon qubits, spinbased qubits, and the like. The superconducting qubits may be located in a cryostat to cool the qubits to superconducting temperatures (e.g., less than about 3 Kelvin). However, aspects of the present disclosure are not limited to superconducting qubits. In some examples, any suitable qubit structure may be used without deviating from the scope of the present disclosure, such as photonic qubits, trapped ion qubits, spin qubits, neutral atom qubits, quantum dot qubits, molecular qubits, or other qubits.

[0027] The type of multi-level quantum subsystems that the system 100 utilizes may vary. For example, in some cases it may be convenient to include one or more readout device(s) 114 attached to one or more superconducting qubits, e g., transmon, flux, gmon, xmon, or other qubits. In other cases, ion traps, photonic devices or superconducting cavities (e.g., with which states may be prepared without requiring qubits) may be used. Further examples of realizations of multi-level quantum subsystems include fluxmon qubits, silicon quantum dots or phosphorus impurity qubits.

[0028] Quantum circuits may be constructed and applied to the register of qubits included in the quantum system 110 via multiple control lines that are coupled to one or more control devices 112. Example control devices 112 that operate on the register of qubits can be used to implement quantum gates or quantum circuits having a plurality of quantum gates, e.g., Pauli gates, Hadamard gates, controlled-NOT (CNOT) gates, controlled-phase gates, T gates, multi-qubit quantum gates, coupler quantum gates, etc. The one or more control devices 112 may be configured to operate on the quantum system 110 through one or more respective control parameters (e.g., one or more physical control parameters). For example, in some implementations, the multi-level quantum subsystems may be superconducting qubits and the control devices 112may be configured to provide control pulses to control lines to generate magnetic fields to adjust the frequency of the qubits.

[0029] The quantum hardware 102 may further include readout devices 114 (e.g., readout resonators). Measurement results 108 obtained via measurement devices may be provided to the classical processors 104 for processing and analyzing. In some implementations, the quantum hardware 102 may include a quantum circuit and the control device(s) 112 and readout devices(s) 114 may implement one or more quantum logic gates that operate on the quantum system 102 through physical control parameters (e.g., microwave pulses) that are sent through wires included in the quantum hardware 102. Further examples of control devices include arbitrary waveform generators, wherein a DAC (digital to analog converter) creates the signal.

[0030] The readout device(s) 114 may be configured to perform quantum measurements on the quantum system 110 and send measurement results 108 to the classical processors 104. In addition, the quantum hardware 102 may be configured to receive data specifying physical control qubit parameter values 106 from the classical processors 104. The quantum hardware 102 may use the received physical control qubit parameter values 106 to update the action of the control device(s) 112 and readout devices(s) 114 on the quantum system 110. For example, the quantum hardware 102 may receive data specifying new values representing voltage strengths of one or more DACs included in the control devices 112 and may update the action of the DACs on the quantum system 110 accordingly. The classical processors 104 may be configured to initialize the quantum system 110 in an initial quantum state, e.g., by sending data to the quantum hardware 102 specifying an initial set of parameters 106.

[0031] In some implementations, the readout device(s) 114 can take advantage of a difference in the impedance for the |0) and |1) states of an element of the quantum system, such as a qubit, to measure the state of the element (e.g., the qubit). For example, the resonance frequency of a readout resonator can take on different values when a qubit is in the state 10) or the state 11), due to the nonlinearity of the qubit. Therefore, a microwave pulse reflected from the readout device 114 carries an amplitude and phase shift that depend on the qubit state. In some implementations, a Purcell filter can be used in conjunction with the readout device(s) 114 to impede microwave propagation at the qubit frequency.

[0032] In some embodiments, the quantum system 110 can include a plurality of qubits 120 arranged, for instance, in a two-dimensional grid 122. For clarity, the two-dimensional grid 122 depicted in FIG. 1 includes 4x4 qubits, however in some implementations the system 110 may include a smaller or a larger number of qubits. In some embodiments, the multiple qubits 120 caninteract with each other through multiple qubit couplers, e.g., qubit coupler 124. The qubit couplers can define nearest neighbor interactions between the multiple qubits 120. In some implementations, the strengths of the multiple qubit couplers are tunable parameters. In some cases, the multiple qubit couplers included in the quantum computing system 100 may be couplers with a fixed coupling strength.

[0033] In some implementations, the multiple qubits 120 may include data qubits, such as qubit 126 and measurement qubits, such as qubit t,. A data qubit is a qubit that participates in a computation being performed by the system 100. A measurement qubit is a qubit that may be used to determine an outcome of a computation performed by the data qubit That is, during a computation an unknown state of the data qubit is transferred to the measurement qubit using a suitable physical operation and measured via a suitable measurement operation performed on the measurement qubit.

[0034] In some implementations, each qubit in the multiple qubits 120 can be operated using respective operating frequencies, such as an idling frequency and / or an interaction frequency and / or readout frequency and / or reset frequency. The operating frequencies can vary from qubit to qubit. For instance, each qubit may idle at a different operating frequency. The operating frequencies for the qubits 120 can be chosen before a computation is performed.

[0035] FIG. 1 depicts one example quantum computing system that can be used to implement the methods and operations according to example aspects of the present disclosure. Other quantum computing systems can be used without deviating from the scope of the present disclosure.Learning Quantum States via Shadow Tomography

[0036] Efficiently extracting information from quantum states (e.g., learning and / or determining the amplitudes of a quantum state encoded in a quantum system) is a central task in quantum computing and quantum information science. More specifically, extracting quantum information encoded in qubits (or other systems encoding quantum states) may be required in physical experiments, simulations run on quantum computing systems, and / or other quantum computation / information processing tasks. In many scenarios, learning the details of an entire quantum state (e.g., the wavefunction of an entangled set of qubits) may not be strictly required for the task at hand. Rather, in many scenarios, learning (or extracting) the expectation values of a set of observables may be sufficient for the task.

[0037] A foundational principle of quantum mechanics is that every observable (e.g., a quantity that is measurable) corresponds to a Hermitian operator. Furthermore, some pairs of such Hermitian operators do not commute Non-commuting operators give rise to another foundational principle of quantum mechanics: the uncertainty principle. The uncertainty principle applies to pairs of non-commuting operators. In one formulation of the uncertainty principle, performing a first measurement associated with a first operator, which does not commute with a second operator, generates some information about the quantum state that is associated with the first operator. However, the first measurement “forces” the quantum state into an eigentstate of the first operator, effectively “collapsing” the wavefunction and “destroying” information that would be (in principle) measurable via a second measurement that is associated with the second operator. Thus, measurements of many properties of quantum states are constrained by the uncertainty principle. This greatly complicates the determination (or learning) of a quantum state of a quantum system.

[0038] Quantum tomography is a set of techniques that, if one is provided with many copies of a quantum state, enables the learning of the quantum state via measuring different observables across the many copies. In general, quantum tomography includes making many noncommuting measurements separately on the many copies of the quantum state. The statistical distributions of the measurements of the non-commuting operators may, at least in principle, allow for the reconstruction of the underlying quantum state. However, tomographic techniques for exactly learning a quantum state up to a stringent standard like worst case observable error (or trace distance) are known to scale exponentially in the number of qubits or polynomially in the size of the Hilbert space. Furthermore, due to the no-cloning theorem, each of the separate copies must be “prepared” because, in contrast to classical information encodable in classical bits, one cannot simply generate multiple copies of a quantum state via a copy operation.

[0039] Due to these limitations, rather than employing traditional quantum tomography, the embodiments employ shadow tomography to learn at least some of the information that characterizes a quantum state of interest. As described in more detail below, shadow tomography leverages classical and quantum resources to compress and store classical information about the quantum state as a "shadow." Via classical processing, the shadow can be used to estimate various properties of the quantum state. That is, the shadow tomography employed by the embodiments includes techniques that allow for efficiently estimating certain properties of a quantum state without requiring full knowledge of the state itself. Various shadow tomography methods (employed by the embodiments) may include randomized measurements and / or classical postprocessing. The randomized measurements may include applying a series of random unitaryoperators to encodings of a quantum state (e.g., where the unitaries are implemented via a quantum circuit). The application of the unitaries may be followed by measurements of the quantum state. The post-processing may include analyzing the collected measurement data to estimate expectation values of observables. In general, shadow tomography requires fewer measurements (e.g., thus less copies of the quantum state) than traditional tomography. Accordingly, shadow tomography scales more efficiently with the dimensionality of the quantum state Furthermore, shadow tomography provides for the simultaneous estimation of multiple (non-commuting) observables. The shadow tomography of the embodiments may employ the concept of classical shadows. Rather than reconstructing the full quantum state via traditional tomography (which requires exponentially many resources and / or copies), classical shadows capture at least some of the essential information about the state required to estimate specific properties or observables. This is done by encoding information about the quantum state into classical data derived from a sequence of the randomized measurements mentioned above. The encoding and packaging of the randomized measurements in a classical dataset may be referred to as a “classical shadow” for the quantum state.

[0040] To summarize the above, via shadow tomography techniques, a set of expectation values of even non-commuting observables may be learned (e.g., estimated and / or determined) within high confidence levels and with a greatly reduced number of samples. For instance, the number of samples required for shadow tomography scales polylogarithmically in the number of observables being measured. While powerful and as noted above, general tomography schemes suffer from two caveats: they are computationally inefficient, and they require immense quantum memories (e.g., millions of times the size of the original state) to enable a large number of entangled measurements. To circumvent these two caveats, the embodiments employ classical shadows. Via classical shadows, extracting observables is computationally efficient, and requires only single-copy measurements for a wide class of useful observables. However, classical shadows place limitations on the sets of observables that are available, for example some schemes are only able to learn observables which are either local or low rank.

[0041] Some of the limitations of classical shadows performed on single copies at a time are fundamental. For general quantum states, certain collections of observables can only be learned with a logarithmic number of samples by exploiting entangled measurements across multiple copies of a state. This is true even for some of the simplest large sets of observables, namely Pauli operators on n qubits. Phrased a different way, the ability to make entangled measurements on copies of a quantum state can provide exponentially more power in learningtasks. As used throughout, the terms “learning” and “learning tasks” may be used to refer to determining and / or estimating information about an unknown quantum state. Such information may include expectation values of various observables of the quantum state. In at least some embodiments, such information may include at least some of the amplitudes (e.g., magnitude and / or phase) of the wavefunction of the quantum state. Such schemes require a quantum memory to simultaneously store copies of the unknown state. This type of advantage cannot be overcome even by an arbitrary amount of classical computation when samples are limited. It has been experimentally demonstrated that this advantage persists even for a small numbers of qubits and in the presence of noise. More specifically, it has been shown that exponential advantages are available using only two copies at a time of the state in quantum memory, or an example of a minimal quantum memory for which the number of copies required is independent of the learning task. This demonstrates the existence of learning tasks for which extremely limited quantum resources (e.g., large numbers of copies of a quantum state, large quantum memories to store the large number of copies, and large numbers of quantum computing resources to manipulate the large number of copies) could provide significant advantages. These advantages of the embodiments are in contrast to the need of traditional tomography that requires memories that are millions of times larger than the system of interest even when tasked with estimating observables to a precision of only 10-3. As such, the embodiments are directed to efficient quantum memories that store very limited numbers of quantum states.

[0042] More particularly, the embodiments are directed towards learning an unknown quantum state of interest via an encoding of the quantum state and an encoding of the conjugate state of the quantum state. The embodiments provide exponential advantages using only a minimal quantum memory (e.g., memories that have enough space to store only 2 copies of an unknown quantum state: a first copy and a copy of its conjugate state). The embodiments also include the ability to make joint measurements on the unknown quantum state with its complex conjugate, denoted as p ® p* . The embodiments enable a quantum learning task that can be achieved with low sample complexity using measurements on p ® p* . In contrast, without access to p*, the same learning task requires exponentially more measurements, unless the size of the quantum memory is allowed to expand to practically unrealistic sizes as a function of the learning task. Quantum memory access to p and p* (e.g., as the embodiments provide) allow for an efficient means to sample in the Heisenberg-Weyl basis, which in turn allows for accomplishing the learning task. As discussed below, a wide class of use cases provide access to the complex conjugate of a quantum state.

[0043] Previous attempts on using minimal quantum memory have only been able to resolve the magnitude of the observable and require a quantum memory scaling polynomially in precision to determine the sign. In contrast, some embodiments may determine both the magnitude and sign using a minimal quantum memory. Below, a specialization for the specific class of operators of interest is discussed.

[0044] The embodiments give rise to an exponential separation that holds for a natural and physically motivated set of observables. The setting is a d-dimensional Hilbert space that discretizes position and momentum space with a natural limit of a continuous bosonic mode. In the infinite dimensional limit, the bosonic displacement operators may be learned. These operators naturally correspond to real -space arrays of quantum sensors. Reconfigurable atom arrays may provide a fruitful test bed for applications in a sensing context for example, especially given their wide bandwidth and sensitivity in other applications. In addition to the learning algorithm using p ® p*, the embodiments include a version of classical shadows tailored to the d-dimensional bosonic setting. It uses a uniform distribution over the generalized d-dimensional Clifford group to make single copy measurements with good predictive power. Despite the more limited power of single copies, the embodiments employ a wide class of quantities that are efficiently learnable.

[0045] One technique of the learning algorithm of the embodiments (e.g., using p ® p*) relies on using extensions via tensor products to create commutativity among displacement operators with Pauli operators as a special case. This technique can be applicable to a broader class of operators than displacement operators. Towards this end, it may be shown that commutativity via tensor extension naturally corresponds to a definition of the Heisenberg-Weyl group with a relationship to uniqueness via the Stone-von Neumann theorem.

[0046] Two applications of the embodiments are herein discussed. The first application is to efficiently learn from the output of quantum computations and / or simulations run on a QCS. Given a quantum algorithm and the explicit quantum circuit that prepares a state of interest p, the conjugate state (e.g., p*) may be prepared by storing the state of interest in a quantum memory and rerunning the simulation, but this time, complex-conjugating each of the gates in the quantum circuit that prepared the quantum state. If enough computational resources are available, the circuit and the complex-conjugated circuit may be run in parallel. The state and the conjugate state may be stored in one or more quantum memories. To realize the exponential advantage of this first application of the embodiments, a factor of two overhead in the size of the quantum computer may be incurred. More specifically, entangled measurements on p ® p* may be performed.

[0047] The second realm of applications of the embodiments includes learning from quantum data collected by one or more quantum sensors, such as but not limited to arrays of sensors for long baseline interferometry. At least some of the operators considered herein are naturally related to regular arrays of quantum sensors arranged in real-space. Some non-limiting examples for obtaining p* are provided below. In this setting, the embodiments may employ techniques that represent a specialized form of mixedness testing that allows exponentially improved signal to noise ratios in determining if a signal is present in high background noise settings for certain classes of states. Potential applications and natural sources of p* are also discussed below.Advantages of Employing Conjugate Quantum States

[0048] Below, some background on the operators and states that the embodiments are directed towards learning is discussed. Furthermore, various theorems showing the exponential power of access to the complex conjugate resource in a minimal quantum memory are also discussed. Shifts in discrete position and momentum space are given by the d-dimensional clock and shift operators, Z and X (defined below), which are generalizations of the qubit Pauli operators to d-dimensions.where m: = e12!'1. The operators naturally map to a ID discrete line in real-space, and can be interchanged via the d-dimensional quantum Fourier transform. Combined position and momentum shifts may be lumped together into displacement operators Dq pdefined asDq,p= ei7iqp / dXqZp(3)

[0049] Definition 1. The displacement amplitudes of a d-dimensional state p are:yq.PTr( ?,pp) (4)

[0050] The displacement operators (e.g., Dq p) may be used to form a basis for quantum states, and there are d2displacement amplitudes. A task considered herein is to estimate all yq pamplitudes to precision E given copies of an unknown quantum state p.

[0051] Task 1. Given access to a quantum state p, estimate all the displacement amplitudes [y«7,p]toprecision e with high probability.

[0052] One result that is relevant to the performance of task l is a sample complexity lower bound showing the minimal size of a conventional quantum memory required to efficiently perform task 1 without access to the resource p* . It can be assumed that K copies of pKare measurable at a time, possibly in entangled bases, but allow no access to p* . Theorem 1 below provides a bound for this result.

[0053] Theorem 1. Let d be the dimension of the Hilbert space. Assume d is prime. Anymeasurements.

[0054] Using a quantum memory that stores pK, performing the learning task whilst consuming a number of copies scaling only as polylog(d) may be difficult (or impossible), even if K grows as large as l / ( 12s). Hence it may be impossible (or at least difficult) to efficiently perform the task with a minimal quantum memory.

[0055] This negative result may lead to the conclusion that quantum memories are not as powerful as discussed herein for physical learning tasks. However, the embodiments employ a resolution to this challenge which provides a subtlety in the power of quantum computing in analyzing quantum data. While a quantum memory containing K states of p 0 'are able to learn the displacement amplitudes of p up to a sign. Not only does the learning algorithm using p 0 p* have logarithmic sample complexity, but the algorithm is simple and computationally efficient. Note that the term “up to a sign” is used to convey that because these are unitary but not always Hermitian operators, they may be complex valued and some, but not all, phase information about that value may be learnable with this procedure. The below theorem 2 can be shown.

[0056] Theorem 2. There is an algorithm which can learn all displacement amplitudes up to a possible minus sign, with precision e, using O(logd / s4) samples. The algorithm makes measurements only on copies of p p* contained in a minimal quantum memory. Moreover, the algorithm is computationally efficient.

[0057] FIG. 2 shows an algorithm 200 for learning all displacement amplitudes up to a possible minus sign, according to various embodiments. That is, algorithm 200 is a non-limiting example of an algorithm whose existence is guaranteed via Theorem 2. More specifically, algorithm 200 allows for the learning of all displacement amplitudes up a possible minus sign up to with precision e, using O(logd / e4) samples. Algorithm 200 makes measurements only on copies of p 0 p* contained in a minimal quantum memory. Moreover, algorithm 200 is computationally efficient. When viewed together, Theorem 1 and Theorem 2 highlight the exponential advantage of using p* as a resource in learning tasks. Indeed, it can be shown that p* is a powerful resource in other contexts, and may not be available for totally general unknown states. Entangled measurements may be a necessary component of the learning algorithm in Theorem 2 (e.g., algorithm 200). This is stated more precisely in Theorem 3.

[0058] Theorem 3. Let d be the dimension of the Hilbert space. Any single-copy protocol which learns the magnitudes of all displacement amplitudes to precision s with probability 2 / 3 requires a number of copies scaling as Cl(d / e2). This holds even if the protocol has access to single-copy measurements of both p and p*.

[0059] Theorem 3 indicates that the conclusion that entangled measurements using quantum memories have dramatically more power than those that can process only a single copy at a time, and indeed that p* inside a minimal quantum memory is a powerful and novel resource.

[0060] An algorithm to learn the displacement amplitudes up to a sign may use a technique where one attaches a second system and constructs a mutually commuting set of operators on the joint system which contain information about the original non-commuting operators on the single system. This technique may be used to develop a shadow tomography algorithm for Pauli operators. The ability to use this trick may be unique to displacement operators which arise from representations of the Heisenberg groups. Theorem 4 may be proved using an idea reminiscent of the Stone-von Neumann theorem.

[0061] Theorem 4. Let U, V be unitaries of finite order d on some Hilbert space H. Suppose U and V do not commute, but we can attach a second Hilbert space H' with unitaries U, V such that U 0 U and V V commute. Then there is some unitary transformation ofH mapping U,V to a direct sum of displacement operators.

[0062] Theorem 4 suggests that to go beyond these classes of observables with efficient shadow tomography, new quantum learning primitives may be employed.Resolving the Signs

[0063] So far the tasks discussed have only pertained to learning expectation values up to a sign. The signs of these operators clearly contain useful information, and hence the embodiments employ additional algorithms to efficiently measure the signs in an information theoretically and computationally efficient sense using only measurements on p 0 p* as well. Theorem 5 provides insight into how the embodiments may learn the sign of the amplitudes as well.

[0064] Theorem 5. There is an algorithm which can learn all d1displacement amplitudes (including their sign) using O(logc / A) samples. The algorithm makes measurements only on copies of p 0 p*. The runtime of the algorithm is poly(t / ,6; ').

[0065] The algorithm in Theorem 5 relies on two key ideas: one is to use a hypothesis state to ‘shift the origin’ of a subsequent magnitude measurement, and the other is to use matrix multiplicative weights as a subroutine to efficiently determine the hypothesis state. FIG. 3A shows an algorithm 300 for determining the signs of displacement amplitudes, according to various embodiments. That is, algorithm 300 is a non-limiting example of an algorithm whose existence is guaranteed via Theorem 5. More specifically, algorithm 300 allows for the learning of the sign of all d2displacement amplitudes. Note that the output of algorithm 200 of FIG. 2 is employed as an input to algorithm 300. Thus, algorithm 300 may call, as a function call, algorithm 200.Furthermore, also note that step 2 of algorithm 300 uses the output of algorithm 320, which is provided in FIG. 3B. FIG. 3B shows an algorithm 320 for finding a hypothesis state, according to various embodiments. The hypothesis state that is determined (and outputted) in algorithm 320 may be a hypothesis for a classical description of a density matrix for the quantum state determined by the embodiments, which is determined in step 2 of algorithm 300 of FIG. 3. Accordingly, algorithm 300 may call algorithm 320, as a function call.Application 1; Quantum Data From Quantum Computation

[0066] As a first application, some embodiments consider cases where an explicit quantum circuit that is capable of preparing a quantum state of interest is known. In such embodiments, a QCS may prepare the state of interest via the known quantum circuit. In other embodiments of this first application, the quantum state of interest may be prepared via a physical experiment.Subsequent to preparing the quantum state (e.g., either through a quantum circuit of a physical experiment), measurements may be performed as discussed throughout. One scenario for this first application is quantum simulation (e.g., via analog quantum computing). In quantum simulation, one aim is to design quantum algorithms that simulate processes found in quantum systems in nature. In these cases, a quantum algorithm prepares the physical quantum state of interest. Task 1(as defined above) indicates that at least in some scenarios, there are advantages to having a quantum algorithm which prepares state p, rather than accessing copies of p through an experimental setup. In particular, there may be natural learning tasks where performing quantum simulation gives a big benefit?

[0067] While other polynomial advantages to having access to the source code (e.g., the quantum source code that defines the quantum circuit that can prepare the state of interest) are known, the learning task here demonstrates an exponential advantage over black box access. In general, it can sometimes be unclear how to access the complex conjugate p* of the state of interest p. However, if a quantum algorithm (or an equivalent circuit) which prepares p is known, the conjugate state may be readily prepared via conjugating the gates of the algorithm.

[0068] More concretely, suppose unitary U (for which an efficient quantum circuit may be implemented) prepares state p via p = Trs(U\0. . . .0 >< 0. . .01 L7+) . Then by complex conjugating each gate in the circuit, the conjugate unitary I may be implemented. Implementing the complex unitary allows for the preparation of the conjugate state via:

[0069] Theorems 1 and 2 then exhibit an exponential cost saving for Task 1 from having white-box access to a quantum algorithm which prepares a quantum state of interest p.Application 2; Quantum Data From Nature

[0070] As a second category of applications, some embodiments are directed towards unknown quantum states collected from nature. For example, these states could be gathered via quantum sensors or transduced from other quantum systems. The ability to learn about unknown state p with exponentially fewer samples using a minimal quantum memory with only K = 2 prompted experimental demonstrations of this idea showing they were robust even with noisy operations.

[0071] In contrast to the collection of qubits case, the operators considered in d-dimensions here connect naturally with quantum sensor arrays in regular spatial arrangements with connections to applications like very long baseline interferometry enabled by quantum communication. The displacement operators are a natural description of discrete position and momentum for real-space arrays, especially in a quantum regime where few excitations are expected and background thermal noise is high.

[0072] For these scenarios, one way to view the results here is as a specialized form of mixedness testing, where for a natural class of signals, exponentially fewer samples are required to detect the presence of the signal when combined with a sea of background noise. For applicationslike detection of radio signals as in nuclear magnetic resonance (NMR), it is common for an infinite temperature background to be quite strong, and this may find applications in that area. These results provide an additional setting for which exponential advantage in signal detection is possible with quantum memory, outside of the existing hierarchies of conventional quantum memory.Generalized Clifford Shadows

[0073] Some embodiments may learn information from single copies in this d-dimensional setting. An effective method for consuming single copies in the case of qubit operators is classical shadows. For instance, some embodiments may operate on a d-level quantum system, by considering the classical shadows associated with random generalized Clifford circuits. The group Cldof generalized Cliffords in d dimensions is the normaliser of the Heisenberg-Weyl group.

[0074] Theorem 6. Let d be prime and let {[ / ) : j E Zd} be the computational basis for Cd.measurements of p. Moreover, the measurements can be done up front, independent of U, ij, and the classical postprocessing to compute estimates is efficient.

[0075] In other words, w the transition elements of p with respect to any and all stabilizer bases may be measured, i.e., {U|j ) : j E Zd} for all U E Cld, using a number of copies that scales logarithmically with d. Cldincludes, for example, the quantum Fourier transform over d dimensions. Classical shadows may be generated that correspond to the uniform distribution over Cld, by evaluating k-fold twirl channels for Cldfor k up to 3. Like the classical shadows associated with n-qubit Clifford circuits, this variance depends on the Hilbert-Schmidt norm of O, but it also has a further dependence on the overlaps of O with displacement operators in a particular way. Because of this additional dependence, the variance may not be small for all low-rank observables; for instance, it scales linearly with d when O = |j)(j|, which is why we only consider off-diagonal elements (i|UtpU[j ) with i j in Theorem 6. Learning all displacement amplitudes using this particular classical shadows procedure may require Q(d / s2) copies of p, which is consistent with our lower bound result in Theorem 3.Methods

[0076] FIG. 4 shows a method 400 for determining an approximation of a quantum state, according to various embodiments. Method 400 may implement at least portions of algorithm 200 of FIG. 2, algorithm 300 of FIG. 3A, and / or algorithm 320 of FIG. 3B.

[0077] Method 400 begins at block 402, where a first set of quantum states is received. Each quantum state of the first set of quantum states is a copy of a first quantum state (e.g., p in algorithm 200). For instance, in algorithm 200, N copies of p are received. The first set of quantum states includes a first copy of a quantum state. At block 404, a second set of quantum states is received. Each quantum state of the second set of quantum states is a copy of a second quantum state. The second quantum state (e.g., p* in algorithm 200) may be conjugate state of the first quantum state. In algorithm 200, N copies of p* are received. The second set of quantum states includes a first copy of the second quantum state. In some embodiments, blocks 402 and 404 may be combined such that N copies of the tensor product quantum state p ® p* are received, where p is an unknown quantum state.

[0078] At block 406, a first set of values is determined (e.g.,in algorithm 200).Each value of the first set of values corresponds to a separate quantum state of the first set of quantum states. Each value of the first set of values is based on a quantum computing system (QCS) measuring a first observable of the quantum state of the first set of quantum states that corresponds to the value of the first set of value (e.g., line 1 in algorithm 200). The first set of values includes a first value (e.g., a1in algorithm 200). At block 408, a second set of values (e.g., {bkyk=in algorithm 200) is determined. Each value of the second set of values corresponds to a separate quantum state of the second set of quantum state. Determining each value of the second set of values is based on the QCS measuring the first observable of the quantum state of the second set of quantum states that corresponds to the value of the second set of values (e.g., line 1 of algorithm 200). The second set of values includes the second value (e.g., in algorithm 200). At block 410, an approximation of the first quantum state is determined based on each value of the first set of values and each value of the second set of values, including the first value and the second value (e g., lines 4-4 of algorithm 200).Additional Embodiments

[0079] One embodiment includes a method for measuring a first quantum state via a quantum computing system (QCS). Note that the method may implement at least portions of algorithm 200 of FIG. 2, algorithm 300 of FIG. 3A, and / or algorithm 320 of FIG. 3B. The methodincludes receiving a first copy of the first quantum state and receiving a first copy of a second quantum state. The second quantum state (e.g., p*) may be a conjugate state of the first quantum state (e.g., ). The first quantum state and the second quantum state may be referenced via a single quantum state that is a tensor product. That is, the first quantum state and the second quantum state may be referenced as the tensor product p ® p*. In some embodiments, the first quantum state and the second quantum state may not be independent quantum states. For instance, the quantum state p ® p* may be an entangled state that is not factorable into a product state of two independent quantum states. The method may further include determining a first value (e g ,inline1 °f algorithm 200) and determining a second value (e.g.,inline 1 of algorithm 200). Determining the first value may be based on measuring, via the QCS, a first observable of the first copy of the first quantum state. Determining the second value may be based on measuring, via the QCS, the first observable of the first copy of the second quantum state. The method may further include determining an approximation of the first quantum state based on the first value and the second value.

[0080] In some embodiments, the method includes receiving a first set of quantum states, wherein each quantum state of the first set of quantum states is a copy of the first quantum state and the first set of quantum states includes the first copy of the first quantum state. The method includes receiving a second set of quantum states, wherein each quantum state of the second set of quantum states is a copy of the second quantum state, which is the conjugate state of the first quantum state, and the second set of quantum states includes the first copy of the second quantum state. The method includes determining a first set of values, wherein each value of the first set of values corresponds to a separate quantum state of the first set of quantum states and is based on measuring, via the QCS, the first observable of the quantum state of the first set of quantum states that corresponds to the value of the first set of values, and the first set of values includes the first value. The method includes determining a second set of values, wherein each value of the second set of values corresponds to a separate quantum state of the second set of quantum state and is based on measuring, via the QCS, the first observable of the quantum state of the second set of quantum states that corresponds to the value of the second set of values, and the second set of values includes the second value. The method includes determining the approximation of the first quantum state based on each value of the first set of values and each value of the second set of values.

[0081] In some embodiments, the first copy of the first quantum state and the first copy of second quantum state are an entangled pair of quantum states. Measuring the first observable ofthe first copy of the first quantum state and measuring the first observable of the first copy of the second quantum state may be performed in a Bell basis. The first observable may be based on a generalized momentum operator and a generalized position operator.

[0082] The method may further include determining components of a first tensor based on the first value and the second value. The method may further include determining components of a second tensor based on the components of the first tensor. The method may further include determining the approximation of the first quantum state based on the components of the second tensor.

[0083] In some embodiments, the approximation of the first quantum state excludes a phase factor of the quantum state. In other embodiments, the approximation of the first quantum state may include a phase factor of the quantum state. Determining the approximation of the first quantum state may be further based on quantum a quantum tomography algorithm. The quantum tomography algorithm may be a shadow tomography algorithm.

[0084] The first copy of the first quantum state and the first copy of the second quantum state may be received from a quantum detector. The quantum detector may be a gravity detector. In other embodiments, the quantum detector is a medical detector.

[0085] In some embodiments, the first copy of the first quantum state and the first copy of the second quantum state may be received by the QCS. The first copy of the first quantum state and the first copy of the second quantum state may be generated by the QCS. The QCS may generate the first copy of the first quantum state and the first copy of the second quantum state in parallel. In other embodiments, the QCS generates the first copy of the first quantum state and the first copy of the second quantum state in series. In some embodiments, the approximation of the first quantum state includes displacement amplitudes up to a sign. In other embodiments, the approximation of the first quantum state includes the sign of at least some of the displacement amplitudes. Determining the approximation of the first quantum state is further based on a set of displacement indices.

[0086] One embodiment includes a quantum computing system (QCS). The quantum computing system includes a quantum processor and one or more memory devices. The quantum processor includes a set of qubits. The one or more memory devices store computer-readable instructions that when executed by the one or more processors cause the one or more processors to perform operations for operating the QCS. The operations may implement at least portions of algorithm 200 of FIG. 2, algorithm 300 of FIG. 3A, and / or algorithm 320 of FIG. 3B. The operations may include receiving a first copy of the first quantum state and receiving a first copy ofa second quantum state. The second quantum state (e.g., *) may be a conjugate state of the first quantum state (e.g., p). The first quantum state and the second quantum state may be referenced via a single quantum state that is a tensor product. That is, the first quantum state and the second quantum state may be referenced as the tensor product p ® p" In some embodiments, the first quantum state and the second quantum state may not be independent quantum states. For instance, the quantum state p ® p* may be an entangled state that is not factorable into a product state of two independent quantum states. The operations may further include determining a first valuein line 1 of algorithm 200) and determining a second value (e.g.,in line 1 of algorithm 200). Determining the first value may be based on measuring, via the QCS, a first observable of the first copy of the first quantum state. Determining the second value may be based on measuring, via the QCS, the first observable of the first copy of the second quantum state. The operations may further include determining an approximation of the first quantum state based on the first value and the second value.

[0087] In some embodiments, the operations include receiving a first set of quantum states, wherein each quantum state of the first set of quantum states is a copy of the first quantum state and the first set of quantum states includes the first copy of the first quantum state. The operations include receiving a second set of quantum states, wherein each quantum state of the second set of quantum states is a copy of the second quantum state, which is the conjugate state of the first quantum state, and the second set of quantum states includes the first copy of the second quantum state. The operations include determining a first set of values, wherein each value of the first set of values corresponds to a separate quantum state of the first set of quantum states and is based on measuring, via the QCS, the first observable of the quantum state of the first set of quantum states that corresponds to the value of the first set of values, and the first set of values includes the first value. The operations include determining a second set of values, wherein each value of the second set of values corresponds to a separate quantum state of the second set of quantum state and is based on measuring, via the QCS, the first observable of the quantum state of the second set of quantum states that corresponds to the value of the second set of values, and the second set of values includes the second value. The operations include determining the approximation of the first quantum state based on each value of the first set of values and each value of the second set of values.

[0088] Implementations of the digital, classical, and / or quantum subject matter and the digital functional operations and quantum operations described in this specification can be implemented in digital electronic circuitry, suitable quantum circuitry or, more generally, quantumcomputational systems, in tangibly-implemented 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 combinations of one or more of them. The term “quantum computing systems” may include, but is not limited to, quantum computers / computing systems, quantum information processing systems, quantum cryptography systems, or quantum simulators.

[0089] Implementations of the digital and / or quantum subject matter described in this specification can be implemented as one or more digital and / or quantum computer programs, i.e., 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, data processing apparatus. 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 / qubit structures, or a combination of one or more of them. Alternatively or in addition, the program instructions can be encoded on an artificially-generated propagated signal that is capable of encoding digital and / or quantum information (e.g., a machine-generated electrical, optical, or electromagnetic signal) that is generated to encode digital and / or quantum information for transmission to suitable receiver apparatus for execution by a data processing apparatus.

[0090] The terms quantum information and quantum data refer to information or data that is carried by, held, or stored in quantum systems, where the smallest non-trivial system is a qubit, i.e., a system that defines the unit of quantum information. It is understood that the term “qubit” encompasses all quantum systems that may be suitably approximated as a two-level system in the corresponding context. Such quantum systems may include multi-level systems, e.g., with two or more levels. By way of example, such systems can include atoms, electrons, photons, ions or superconducting qubits. In many implementations the computational basis states are identified with the ground and first excited states, however it is understood that other setups where the computational states are identified with higher level excited states (e.g., qubits) are possible.

[0091] The term “data processing apparatus” refers to digital and / or quantum data processing hardware and encompasses all kinds of apparatus, devices, and machines for processing digital and / or quantum data, including by way of example a programmable digital processor, a programmable quantum processor, a digital computer, a quantum computer, or multiple digital and quantum processors or computers, and combinations thereof. The apparatus can also be, or further include, special purpose logic circuitry, e.g., an FPGA (field programmable gate array), or an ASIC(application-specific integrated circuit), or a quantum simulator, i.e., a quantum data processing apparatus that is designed to simulate or produce information about a specific quantum system. In particular, a quantum simulator is a special purpose quantum computer that does not have the capability to perform universal quantum computation. The apparatus can optionally include, in addition to hardware, 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 them.

[0092] A digital or classical computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, or code, can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can 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 digital computing environment. A quantum computer program, which may also be referred to or described as a program, software, a software application, a module, a software module, a script, 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, Quipper, Cirq, etc..

[0093] A digital and / or quantum computer program may, but need not, correspond to a file in a file system. A program can be stored in a portion 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 coordinated files, e.g., files that store one or more modules, sub-programs, or portions of code. A digital and / or quantum computer program can be deployed to be executed on one digital or one quantum computer or on multiple digital and / or quantum computers that are 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, however a quantum data communication network may transmit both quantum data and digital data.

[0094] The processes and logic flows described in this specification can be performed by one or more programmable digital and / or quantum computers, operating with one or more digital and / or quantum processors, as appropriate, executing one or more digital and / or quantum computer programs to perform functions by operating on input digital and quantum data and generating output. The processes and logic flows can also be performed by, and apparatus can also beimplemented as, special purpose logic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or by a combination of special purpose logic circuitry or quantum simulators and one or more programmed digital and / or quantum computers.

[0095] For a system of one or more digital and / or quantum computers or processors to be “configured to” or “operable to” perform particular operations or actions means that the system has installed on it software, firmware, hardware, or a combination of them that in operation cause the system to perform the operations or actions. For one or more digital and / or quantum computer programs to be configured to perform particular operations or actions means that the one or more programs include instructions that, when executed by digital and / or quantum data processing apparatus, cause the apparatus to perform the operations or actions. A quantum computer may receive instructions from a digital computer that, when executed by the quantum computing apparatus, cause the apparatus to perform the operations or actions.

[0096] Digital and / or quantum computers suitable for the execution of a digital and / or quantum computer program can be based on general or special purpose digital and / or quantum microprocessors or both, or any other kind of central digital and / or quantum processing unit. Generally, a central digital and / or quantum processing unit will receive instructions and digital and / or quantum data from a read-only memory, or a random access memory, or quantum systems suitable for transmitting quantum data, e.g. photons, or combinations thereof.

[0097] Some example 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 the memory can be supplemented by, or incorporated in, special purpose logic circuitry or quantum simulators. Generally, a digital and / or quantum computer will also include, or be operatively coupled to receive digital and / or quantum data from or transfer digital and / or quantum data to, or both, one or more mass storage devices for storing digital and / or quantum data, e.g., magnetic, magneto-optical disks, or optical disks, or quantum systems suitable for storing quantum information. However, a digital and / or quantum computer need not have such devices.

[0098] 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 nonvolatile digital and / or quantum memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto-optical disks; and CD-ROM and DVD- ROM disks; and quantum systems, e.g., trapped atoms or electrons. It is understood that quantummemories are devices that can store quantum data for a long time with high fidelity and efficiency, e.g., light-matter interfaces where light is used for transmission and matter for storing and preserving the quantum features of quantum data such as superposition or quantum coherence.

[0099] Control of the various systems described in this specification, or portions of them, can be implemented in a digital and / or quantum computer program product that includes instructions that are stored on one or more tangible, non-transitory machine-readable storage media, and that are executable on one or more digital and / or quantum processing devices. The systems described in this specification, or portions of them, can each be implemented as an apparatus, method, or electronic system that may include one or more digital and / or quantum processing devices and memory to store executable instructions to perform the operations described in this specification.

[0100] While this specification contains many specific implementation details, 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 particular implementations. Certain features that are described in this specification in the context of separate implementations can also be implemented in combination in a single implementation. Conversely, various features that are described in the context of a single implementation can also be implemented in multiple implementations separately or in any suitable sub combination. Moreover, although 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 excised from the combination, and the claimed combination may be directed to a sub-combination or variation of a sub-combination.

[0101] Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. In certain circumstances, multitasking and parallel processing may be advantageous. Moreover, 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 described program components and systems can generally be integrated together in a single software product or packaged into multiple software products.

[0102] Particular implementations of the subject matter have been described. Other implementations are within the scope of the following claims. For example, the actions recited in the claims can be performed in a different order and still achieve desirable results. As one example, the processes depicted in the accompanying figures do not necessarily require the particular orderT1shown, or sequential order, to achieve desirable results. In some cases, multitasking and parallel processing may be advantageous.

Claims

WHAT IS CLAIMED IS:

1. A method for measuring a first quantum state via a quantum computing system (QCS), the method comprising: receiving a first copy of the first quantum state; receiving a first copy of a second quantum state, wherein the second quantum state is a conjugate state of the first quantum state; determining a first value based on measuring, via the QCS, a first observable of the first copy of the first quantum state; determining a second value based on measuring, via the QCS, the first observable of the first copy of the second quantum state; and determining an approximation of the first quantum state based on the first value and the second value.

2. The method of claim 1, further comprising: receiving a first set of quantum states, wherein each quantum state of the first set of quantum states is a copy of the first quantum state and the first set of quantum states includes the first copy of the first quantum state; receiving a second set of quantum states, wherein each quantum state of the second set of quantum states is a copy of the second quantum state, which is the conjugate state of the first quantum state, and the second set of quantum states includes the first copy of the second quantum state; determining a first set of values, wherein each value of the first set of values corresponds to a separate quantum state of the first set of quantum states and is based on measuring, via the QCS, the first observable of the quantum state of the first set of quantum states that corresponds to the value of the first set of values, and the first set of values includes the first value; determining a second set of values, wherein each value of the second set of values corresponds to a separate quantum state of the second set of quantum state and is based on measuring, via the QCS, the first observable of the quantum state of the second set of quantum states that corresponds to the value of the second set of values, and the second set of values includes the second value; and determining the approximation of the first quantum state based on each value of the first set of values and each value of the second set of values.

3. The method of claim 1, wherein the first copy of the first quantum state and the first copy of second quantum state are an entangled pair of quantum states.

4. The method of claim 3, wherein measuring the first observable of the first copy of the first quantum state and measuring the first observable of the first copy of the second quantum state is performed in a Bell basis.

5. The method of claim 1, wherein the first observable is based on a generalized momentum operator and a generalized position operator.

6. The method of claim 1, further comprising: determining components of a first tensor based on the first value and the second value; determining components of a second tensor based on the components of the first tensor; and determining the approximation of the first quantum state based on the components of the second tensor.

7. The method of claim 1, wherein the approximation of the first quantum state excludes a phase factor of the quantum state.

8. The method of claim 1, wherein determining the approximation of the first quantum state is further based on quantum a quantum tomography algorithm.

9. The method of claim 8, wherein the quantum tomography algorithm is a shadow tomography algorithm.

10. The method of claim 1, wherein the first copy of the first quantum state and the first copy of the second quantum state are received from a quantum detector.

11. The method of claim 10, wherein the quantum detector is a gravity detector.

12. The method of claim 10, wherein the quantum detector is a medical detector.

13. The method of claim 1, wherein the first copy of the first quantum state and the first copy of the second quantum state are received by the QCS.

14. The method of claim 1, wherein the first copy of the first quantum state and the first copy of the second quantum state are generated by the QCS.

15. The method of claim 14, wherein the QCS generates the first copy of the first quantum state and the first copy of the second quantum state in parallel.

16. The method of claim 14, wherein the QCS generates the first copy of the first quantum state and the first copy of the second quantum state in series.

17. The method of claim 1, wherein the approximation of the first quantum state includes displacement amplitudes up to a sign.

18. The method of claim 1, wherein determining the approximation of the first quantum state is further based on a set of displacement indices.

19. A quantum computing system (QCS) comprising: a quantum processor that includes a set of qubits; one or more memory devices, the one or more memory devices storing computer- readable instructions that when executed by the one or more processors cause the one or more processors to perform operations for operating the QCS, the operations comprising: receiving a first copy of the first quantum state; receiving a first copy of a second quantum state, wherein the second quantum state is a conjugate state of the first quantum state; determining a first value based on measuring a first observable of the first copy of the first quantum state; determining a second value based on measuring the first observable of the first copy of the second quantum state; anddetermining an approximation of the first quantum state based on the first value and the second value.

20. The QCS of claim 19, wherein the operations further comprise: receiving a first set of quantum states, wherein each quantum state of the first set of quantum states is a copy of the first quantum state and the first set of quantum states includes the first copy of the first quantum state; receiving a second set of quantum states, wherein each quantum state of the second set of quantum states is a copy of the second quantum state, which is the conjugate state of the first quantum state, and the second set of quantum states includes the first copy of the second quantum state; determining a first set of values, wherein each value of the first set of values corresponds to a separate quantum state of the first set of quantum states and is based on measuring the first observable of the quantum state of the first set of quantum states that corresponds to the value of the first set of values, and the first set of values includes the first value; determining a second set of values, wherein each value of the second set of values corresponds to a separate quantum state of the second set of quantum state and is based on measuring the first observable of the quantum state of the second set of quantum states that corresponds to the value of the second set of values, and the second set of values includes the second value; and determining the approximation of the first quantum state based on each value of the first set of values and each value of the second set of values

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