ABSTRACT
Abstract
Methods, systems, and apparatus for quantum machine learning. In one aspect, a method includes obtaining, by a quantum computing device, a training dataset of quantum data points; computing, by the quantum computing device, a kernel matrix that represents a similarity between the quantum data points included in the training dataset, comprising computing a value of a kernel function for each pair of quantum data points in the training dataset, wherein the kernel function is based on reduced density matrices for the quantum data points; and providing, by the quantum computing device, the kernel matrix to a classical processor, wherein the classical processor performs a training algorithm using the kernel matrix to construct a machine learning model.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims the benefit under 35 U.S.C. § 119(e) of U.S. Patent Application No. 63/093,611, filed Oct. 19, 2020. The disclosure of the foregoing application is incorporated herein by reference in its entirety for all purposes.
BACKGROUND
Kernel methods are a class of algorithms for pattern analysis. The task of pattern analysis is to find and study general types of relations, e.g., clusters, rankings, principal components, correlations, and classifications in datasets. For many algorithms that solve these tasks, the data in raw representation has to be explicitly transformed into feature vector representations via a user-specified feature map. In contrast, kernel methods require only a user-specified kernelâa similarity function (or âkernel functionâ) over pairs of data points in raw representation.
Kernel functions enable kernel methods to operate in a high-dimensional, implicit feature space without computing the coordinates of the data in that space. Instead, inner products between images of all pairs of data in the feature space are computed. These operations are often computationally cheaper than the explicit computation of the coordinates.
Algorithms capable of operating with kernels include the kernel perceptron, support vector machines (SVM), Gaussian processes, principal components analysis (PCA), canonical correlation analysis, ridge regression, spectral clustering, linear adaptive filters and many others. Any linear model can be turned into a non-linear model by applying the kernel trick to the model: replacing its features (predictors) by a kernel function.
SUMMARY
This specification describes techniques for quantum computing with kernel methods for machine learning.
In general, one innovative aspect of the subject matter described in this specification can be implemented in methods that include obtaining, by a quantum computing device, a training dataset of quantum data points; computing, by the quantum computing device, a kernel matrix that represents similarities among the quantum data points included in the training dataset, comprising computing, for each pair of quantum data points in the training dataset, a corresponding value of a kernel function, wherein the kernel function is based on reduced density matrices for the quantum data points; and providing, by the quantum computing device, the kernel matrix to a classical processor.
Other implementations of this aspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions.
The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations the method further comprises receiving, from the quantum computing device and by the classical processor, the kernel matrix; and performing, by the classical processor, a training algorithm using the kernel matrix to construct a machine learning model.
In some implementations the method further comprises obtaining, by the quantum computing device, a validation dataset of quantum data points; computing, by the quantum computing device, new elements of the kernel matrix, wherein the new elements comprise entries representing similarities between the quantum data points in the validation dataset and the quantum data points in the training dataset, wherein computing the new elements comprises computing, for each pair of quantum data points in the training dataset and the validation dataset, corresponding values of the kernel function; and providing, by the quantum computing device, the new elements of the kernel matrix to the classical processor.
In some implementations the method further comprises processing, by the classical processor, the new elements of the kernel matrix to output predictions for each quantum data point in the validation dataset.
In some implementations the kernel function is based on single-body reduced density matrices for the quantum data points in the training dataset.
In some implementations the kernel function comprises a linear kernel function.
In some implementations the linear kernel function i) takes a first quantum data point and second quantum data point as input, ii) produces a numerical output, and iii) includes a sum of terms, wherein the sum runs over each of N-qubits for N>1 and the summand corresponds to a respective qubit and is equal to the trace of a product of a) a reduced density matrix for the first quantum data point on a subsystem corresponding to the respective qubit b) a reduced density matrix for the second quantum data point on a subsystem corresponding to the respective qubit.
In some implementations the linear kernel function is given by
Q â¡
(
x i
,
x j
)
=
â l
â¢
â¢
Tr â¡
[
Tr
m â l
â¡
[
Ï â¡
(
x i
)
]
]
â¡
[
Tr
n â l
â¡
[
Ï â¡
(
x j
)
]
]
where x i , x j represent a first and second quantum data point, l represents an index that runs from 1 to the number of qubits N and labels each qubit, Ï(x i )=|x i
x i | and Tr mâ k [Ï(x i )] represents a 1-reduced density matrix (RDM) on qubit k.
In some implementations computing a value of the kernel function for a pair of quantum data points in the training dataset, the pair comprising a first N-qubit quantum state worth N>1 and a second N-qubit quantum state with N>1, comprises: repeatedly and for each qubit index: computing a 1-reduced density matrix (RDM) for the first N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the first N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a first reduced quantum state of the quantum system; computing a 1-RDM for the second N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the second N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a second reduced quantum state of the quantum system; determining a trace of the product of the first reduced quantum state and the second reduced quantum state; and summing averages of the determined traces for each qubit index.
In some implementations the kernel function comprises a squared exponential kernel function.
In some implementations the squared exponential kernel function i) takes a first quantum data point and second quantum data point as input, ii) produces a numerical output, and iii) includes an exponential function of a sum of terms, wherein the sum runs over each of N-qubits for N>1 and the summand corresponds to a respective qubit and is equal to norm of a) a reduced density matrix for the first quantum data point on a subsystem corresponding to the respective qubit minus b) a reduced density matrix for the second quantum data point on a subsystem corresponding to the respective qubit.
In some implementations the squared exponential kernel function is given by
Q E 1
â¡
(
x i
,
x j
)
=
exp (
- γ
â¢
â l
â¢
ï
Tr
m â l
â¡
[
Ï â¡
(
x i
)
]
-
Tr
n â l
â¡
[
Ï â¡
(
x j
)
]
ï
2
)
where x i , x j represent a first and second quantum data point, l represents an index that runs from 1 to the number of qubits N and labels each qubit, Ï(x i )=|x i
x i | and Tr mâ k [Ï(x i )] represents a 1-RDM on qubit k.
In some implementations computing a value of the kernel function for a pair of quantum data points in the training dataset, the pair comprising a first N-qubit quantum state with N>1 and a second N-qubit quantum state with N>1, comprises repeatedly and for each qubit index: computing a 1-reduced density matrix (RDM) for the first N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the first N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a first reduced quantum state of the quantum system; computing a 1-RDM for the second N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the second N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a second reduced quantum state of the quantum system; subtracting the second reduced quantum state from the first reduced quantum state to obtain a third reduced quantum state and determining a norm of the third reduced quantum state; and summing averages of the determined norms for each qubit index and computing an exponent of the summed averages.
In some implementations the kernel function is based on k-body RDMs for the quantum data points, wherein k is less than a predetermined value.
In some implementations the kernel function comprises a linear kernel function.
In some implementations the linear kernel function i) takes a first quantum data point and second quantum data point as input, ii) produces a numerical output, and iii) comprises a sum of terms, wherein the sum runs over each subset of k-qubits taken from the N qubits and each summand corresponds to a respective subset and is equal to the trace of a product of a) a reduced density matrix for the first quantum data point on a subsystem corresponding to the respective subset of k qubits and b) a reduced density matrix for the second quantum data point on a subsystem corresponding to the respective subset of k qubits.
In some implementations the linear kernel function is given by
Q L k
â¡
(
x i
,
x j
)
=
â
K â
S k
â¡
( n )
â¢
â¢
Tr â¡
[
Tr
m â K
â¡
[
Ï â¡
(
x i
)
]
]
â¡
[
Tr
n â K
â¡
[
Ï â¡
(
x j
)
]
]
where S k (n) represents a set of subsets of k qubits, Ï(x i )=|x i
<img id="CUSTOM-CHARACTER-00006" he="3.22mm" wi="1.10mm" file="US12321838-20250603-P00002.TIF" alt="Figure US12321838-20250603-P00002" img-content="character" img-format="tif" orientation="portrait" inline="no" width="4" height="13" alt="Figure US12321838-20250603-P00002" class="patent-full-image" src="https://patentimages
CROSS-REFERENCE TO RELATED APPLICATIONS
This application claims the benefit under 35 U.S.C. § 119(e) of U.S. Patent Application No. 63/093,611, filed Oct. 19, 2020. The disclosure of the foregoing application is incorporated herein by reference in its entirety for all purposes.
BACKGROUND
Kernel methods are a class of algorithms for pattern analysis. The task of pattern analysis is to find and study general types of relations, e.g., clusters, rankings, principal components, correlations, and classifications in datasets. For many algorithms that solve these tasks, the data in raw representation has to be explicitly transformed into feature vector representations via a user-specified feature map. In contrast, kernel methods require only a user-specified kernelâa similarity function (or âkernel functionâ) over pairs of data points in raw representation.
Kernel functions enable kernel methods to operate in a high-dimensional, implicit feature space without computing the coordinates of the data in that space. Instead, inner products between images of all pairs of data in the feature space are computed. These operations are often computationally cheaper than the explicit computation of the coordinates.
Algorithms capable of operating with kernels include the kernel perceptron, support vector machines (SVM), Gaussian processes, principal components analysis (PCA), canonical correlation analysis, ridge regression, spectral clustering, linear adaptive filters and many others. Any linear model can be turned into a non-linear model by applying the kernel trick to the model: replacing its features (predictors) by a kernel function.
SUMMARY
This specification describes techniques for quantum computing with kernel methods for machine learning.
In general, one innovative aspect of the subject matter described in this specification can be implemented in methods that include obtaining, by a quantum computing device, a training dataset of quantum data points; computing, by the quantum computing device, a kernel matrix that represents similarities among the quantum data points included in the training dataset, comprising computing, for each pair of quantum data points in the training dataset, a corresponding value of a kernel function, wherein the kernel function is based on reduced density matrices for the quantum data points; and providing, by the quantum computing device, the kernel matrix to a classical processor.
Other implementations of this aspect include corresponding computer systems, apparatus, and computer programs recorded on one or more computer storage devices, each configured to perform the actions of the methods. A system of one or more computers can be configured to perform particular operations or actions by virtue of having software, firmware, hardware, or a combination thereof installed on the system that in operation causes or cause the system to perform the actions. One or more computer programs can be configured to perform particular operations or actions by virtue of including instructions that, when executed by data processing apparatus, cause the apparatus to perform the actions.
The foregoing and other implementations can each optionally include one or more of the following features, alone or in combination. In some implementations the method further comprises receiving, from the quantum computing device and by the classical processor, the kernel matrix; and performing, by the classical processor, a training algorithm using the kernel matrix to construct a machine learning model.
In some implementations the method further comprises obtaining, by the quantum computing device, a validation dataset of quantum data points; computing, by the quantum computing device, new elements of the kernel matrix, wherein the new elements comprise entries representing similarities between the quantum data points in the validation dataset and the quantum data points in the training dataset, wherein computing the new elements comprises computing, for each pair of quantum data points in the training dataset and the validation dataset, corresponding values of the kernel function; and providing, by the quantum computing device, the new elements of the kernel matrix to the classical processor.
In some implementations the method further comprises processing, by the classical processor, the new elements of the kernel matrix to output predictions for each quantum data point in the validation dataset.
In some implementations the kernel function is based on single-body reduced density matrices for the quantum data points in the training dataset.
In some implementations the kernel function comprises a linear kernel function.
In some implementations the linear kernel function i) takes a first quantum data point and second quantum data point as input, ii) produces a numerical output, and iii) includes a sum of terms, wherein the sum runs over each of N-qubits for N>1 and the summand corresponds to a respective qubit and is equal to the trace of a product of a) a reduced density matrix for the first quantum data point on a subsystem corresponding to the respective qubit b) a reduced density matrix for the second quantum data point on a subsystem corresponding to the respective qubit.
In some implementations the linear kernel function is given by
Q â¡
(
x i
,
x j
)
=
â l
â¢
â¢
Tr â¡
[
Tr
m â l
â¡
[
Ï â¡
(
x i
)
]
]
â¡
[
Tr
n â l
â¡
[
Ï â¡
(
x j
)
]
]
where x i , x j represent a first and second quantum data point, l represents an index that runs from 1 to the number of qubits N and labels each qubit, Ï(x i )=|x i
x i | and Tr mâ k [Ï(x i )] represents a 1-reduced density matrix (RDM) on qubit k.
In some implementations computing a value of the kernel function for a pair of quantum data points in the training dataset, the pair comprising a first N-qubit quantum state worth N>1 and a second N-qubit quantum state with N>1, comprises: repeatedly and for each qubit index: computing a 1-reduced density matrix (RDM) for the first N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the first N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a first reduced quantum state of the quantum system; computing a 1-RDM for the second N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the second N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a second reduced quantum state of the quantum system; determining a trace of the product of the first reduced quantum state and the second reduced quantum state; and summing averages of the determined traces for each qubit index.
In some implementations the kernel function comprises a squared exponential kernel function.
In some implementations the squared exponential kernel function i) takes a first quantum data point and second quantum data point as input, ii) produces a numerical output, and iii) includes an exponential function of a sum of terms, wherein the sum runs over each of N-qubits for N>1 and the summand corresponds to a respective qubit and is equal to norm of a) a reduced density matrix for the first quantum data point on a subsystem corresponding to the respective qubit minus b) a reduced density matrix for the second quantum data point on a subsystem corresponding to the respective qubit.
In some implementations the squared exponential kernel function is given by
Q E 1
â¡
(
x i
,
x j
)
=
exp (
- γ
â¢
â l
â¢
ï
Tr
m â l
â¡
[
Ï â¡
(
x i
)
]
-
Tr
n â l
â¡
[
Ï â¡
(
x j
)
]
ï
2
)
where x i , x j represent a first and second quantum data point, l represents an index that runs from 1 to the number of qubits N and labels each qubit, Ï(x i )=|x i
x i | and Tr mâ k [Ï(x i )] represents a 1-RDM on qubit k.
In some implementations computing a value of the kernel function for a pair of quantum data points in the training dataset, the pair comprising a first N-qubit quantum state with N>1 and a second N-qubit quantum state with N>1, comprises repeatedly and for each qubit index: computing a 1-reduced density matrix (RDM) for the first N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the first N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a first reduced quantum state of the quantum system; computing a 1-RDM for the second N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the second N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a second reduced quantum state of the quantum system; subtracting the second reduced quantum state from the first reduced quantum state to obtain a third reduced quantum state and determining a norm of the third reduced quantum state; and summing averages of the determined norms for each qubit index and computing an exponent of the summed averages.
In some implementations the kernel function is based on k-body RDMs for the quantum data points, wherein k is less than a predetermined value.
In some implementations the kernel function comprises a linear kernel function.
In some implementations the linear kernel function i) takes a first quantum data point and second quantum data point as input, ii) produces a numerical output, and iii) comprises a sum of terms, wherein the sum runs over each subset of k-qubits taken from the N qubits and each summand corresponds to a respective subset and is equal to the trace of a product of a) a reduced density matrix for the first quantum data point on a subsystem corresponding to the respective subset of k qubits and b) a reduced density matrix for the second quantum data point on a subsystem corresponding to the respective subset of k qubits.
In some implementations the linear kernel function is given by
Q L k
â¡
(
x i
,
x j
)
=
â
K â
S k
â¡
( n )
â¢
â¢
Tr â¡
[
Tr
m â K
â¡
[
Ï â¡
(
x i
)
]
]
â¡
[
Tr
n â K
â¡
[
Ï â¡
(
x j
)
]
]
where S k (n) represents a set of subsets of k qubits, Ï(x i )=|x i
x i | and Tr mâ K [Ï(x i )] represents a k-RDM.
In some implementations computing a value of the kernel function for a pair of quantum data points in the training dataset, the pair comprising a first N-qubit quantum state and a second N-qubit quantum state, comprises: repeatedly and for each set of k-qubits: computing a k-RDM for the first N-qubit quantum state on a subsystem corresponding to qubits in the set, comprising obtaining a copy of an N-qubit quantum system in the first N-qubit quantum state and measuring each qubit in the quantum system except for the qubits included in the set to obtain a first reduced quantum state of the quantum system; computing a k-RDM for the second N-qubit quantum state on a subsystem corresponding to qubits in the set, comprising obtaining a copy of an N-qubit quantum system in the second N-qubit quantum state and measuring each qubit in the quantum system except for the qubits included in the set to obtain a second reduced quantum state of the quantum system; determining a trace of the product of the first reduced quantum state and the second reduced quantum state; and summing averages of the determined for each set of k qubits.
In some implementations the kernel function comprises an exponential kernel function.
In some implementations the exponential kernel function is given by
Q S
â¡
(
x i
,
x j
)
=
â
k = 0
â
â¢
â¢
γ k
k !
â¢
n k
â¢
Q l k
â¡
(
x i
,
x j
)
=
ð¼ â¢
â¢
exp â¡
(
γ n
â¢
â
h = 1
n
â¢
â¢
(
9 â¢
δ
s h i
â¢
s h j
â¢
δ
b h i
â¢
b h j
- 4
)
)
where the expected value
is taken over n s samples (i.e., experiments, where n s is chosen as large as possible whilst taking into account hardware implementation considerations, e.g., n s is chosen as the largest number that can be afforded to be measured experimentally), from randomly chosen Pauli frames that are measured on a first and second system i and j,
δ
s h i
â¢
s h j
represents a first indicator function for an agreement between a random Pauli measurement result performed independently on the first system i and the second system j and
δ
b h i
â¢
b h j
represents a second indicator function for a measurement basis agreement.
In some implementations computing a value of the kernel function for a pair of quantum data points in the training dataset, the pair comprising a first N-qubit quantum state and a second N-qubit quantum state, comprises repeatedly: obtaining a first measurement result, comprising measuring each qubit in the first system in a random Pauli basis to obtain values s h i and b h i for the h-th qubit, wherein s h i is either 1 or â1 and b h i is a random basis X, Y, or Z; obtaining a second measurement result, comprising measuring each qubit in the second system in a random Pauli basis to obtain values s h j and b h j for the h-th qubit, wherein s h j is either 1 or â1 and b h j is a random basis X, Y, or Z; comparing, for the h-th qubit in the N qubit system, the first measurement result and second measurement result to determine a value of the first indicator function, determining, for the h-th qubit in the N qubit system, a value of the second indicator function; and multiplying, summing and averaging the determined values of the first indicator function and the second indicator function.
In some implementations the quantum data points comprise N-qubit quantum states with N>1.
In some implementations obtaining the training dataset of quantum data points comprises: receiving a training dataset of classical data points; and generating the training dataset of quantum data points, comprising embedding each classical data point in a respective quantum state by applying a respective encoding circuit to a reference quantum state.
The subject matter described in this specification can be implemented in particular ways so as to realize one or more of the following advantages.
Kernel methods for machine learning can be applied to a variety of regression and classification problems. However, there are limitations to the successful solution to such problems when the feature space becomes large and the kernel function becomes computationally expensive to estimate. The presently described techniques address this problem by using a quantum computing device to compute the kernel function.
In addition, the presently described quantum computation of the kernel function is scalableâas the number of qubits increases, the signal remains large and the method continues to function well, if not better. This is in contrast to known quantum kernel methods where the signal typically decays exponentially in the number of qubits, e.g., because of a small geometric difference due to an exponentially large Hilbert space where all inputs are too far apart. The scalability of the presently described techniques is achieved by enlarging the geometric difference by projecting quantum states embedded from classical data back to classical space, e.g., through using RDMs. In other words, a kernel function that is close to zero for every two points does not generalize well. However, the presently described projected quantum kernel is defined using an approximate classical representation of the quantum state, and this results in a non-zero kernel function that provides better generalization performance.
Further, due to the enlarged geometric difference, the presently described techniques can achieve a large prediction advantage over common classical machine learning models. Such prediction advantages can also be achieved with a small number of qubits, e.g., up to 30 qubits. Therefore, the presently described techniques are particularly suitable for implementations using small quantum computers, e.g., noisy intermediate scale quantum devices and/or hybrid quantum-classical computers.
The presently described techniques can be applied to various applications of classical machine learning, including examples from quantum machine learning that involve naturally quantum input data, including: image and digit classification such as from MNIST or other sources of image/video data, classification of sentiment and textual analysis, analysis of high energy physics data, classification of data from a quantum sensor into a phase, quantum state discrimination or quantum repeater engineering, prediction using data from quantum sensors, many-body or otherwise.
The details of one or more implementations of the subject matter of this specification 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, the drawings, and the claims.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is an illustration of kernel functions defined by classical kernel methods, conventional quantum kernel methods, and the projected quantum kernel method.
FIG. 2 shows a block diagram of an example system for performing classification and regression tasks using a projected quantum kernel method.
FIG. 3 shows a block diagram of an example process for performing classification and regression tasks using a projected quantum kernel method.
FIG. 4 is a flowchart of an example process for generating and updating a kernel matrix.
Like reference numbers and designations in the various drawings indicate like elements.
DETAILED DESCRIPTION
This specification describes techniques for performing machine learning tasks using a quantum kernel method.
In conventional quantum kernel methods the kernel operator is based on a fidelity-type metric, e.g., given by Tr[Ï(x i )Ï(x j )]. This kernel operator can regard all data points to be far from each other and produce a kernel matrix that is close to identity. This can result in a small geometric difference and can lead to classical machine learning models being competitive or outperforming the quantum kernel method. For example, in some cases a quantum model may require an exponential amount of samples to learn using this conventional kernel operator, but only needs a linear number of samples to learn using a classical machine learning model.
The presently described quantum kernel method addresses this problem using a family of projected quantum kernels. A quantum or classical dataset of data points is received and a quantum computer is used to compute the geometry between the data points. The geometry is computed using a projected quantum kernel operator chosen from a family of reduced physical observables that are scalable. The projected quantum kernel operator projects quantum states to an approximate classical representation, e.g., using reduced observables or classical shadows. The computed geometry is then fed to a classical method for training and verification. Even if the training set space has a large dimension, e.g., a dimension proportional to the number of qubits included in the available quantum computer, the projection provides a reduction to a low-dimensional classical space that can generalize better.
FIG. 1 is an illustration of the geometry (kernel function) defined by classical kernel methods 100 , conventional quantum kernel methods 102 and the presently described projected quantum kernel method 104 . The letters A, B, C, represent data points in different spaces with arrows representing the similarity measure (kernel function) between data. The geometric difference g is a difference between similarity measures in the
different methods
100 , 102 , and 104 and d is an effective dimension of the data set in the quantum Hilbert space. As shown, the geometric difference between the similarity measures in the classical kernel method 100 and the projected quantum kernel method 104 is larger than the geometric difference between the similarity measures in the classical kernel method 100 and the conventional quantum kernel method 102 . This larger geometric difference provides scalability and improved prediction accuracy, as discussed above.
Example Operating Environment
FIG. 2 depicts an example system 200 for performing classification and regression tasks using a projected quantum kernel method. The example system 200 is an example of a system implemented as classical and quantum computer programs on one or more classical computers and quantum computing devices in one or more locations, in which the systems, components, and techniques described below can be implemented.
The example system 200 includes an example quantum computing device 202 . The quantum computing device 202 can be used to perform the quantum computation operations described in this specification according to some implementations. The quantum computing device 202 is intended to represent various forms of quantum computing devices. The components shown here, their connections and relationships, and their functions, are exemplary only, and do not limit implementations of the inventions described and/or claimed in this document.
The example quantum computing device 202 includes a qubit assembly 252 and a control and measurement system 204 . The qubit assembly includes multiple qubits, e.g., qubit 206 , that are used to perform algorithmic operations or quantum computations. While the qubits shown in FIG. 2 are arranged in a rectangular array, this is a schematic depiction and is not intended to be limiting. The qubit assembly 252 also includes adjustable coupling elements, e.g., coupler 208 , that allow for interactions between coupled qubits. In the schematic depiction of FIG. 2 , each qubit is adjustably coupled to each of its four adjacent qubits by means of respective coupling elements. However, this is an example arrangement of qubits and couplers and other arrangements are possible, including arrangements that are non-rectangular, arrangements that allow for coupling between non-adjacent qubits, and arrangements that include adjustable coupling between more than two qubits.
Each qubit can be a physical two-level quantum system or device having levels representing logical values of 0 and 1. The specific physical realization of the multiple qubits and how they interact with one another is dependent on a variety of factors including the type of the quantum computing device included in example system 200 or the type of quantum computations that the quantum computing device is performing. For example, in an atomic quantum computer the qubits may be realized via atomic, molecular or solid-state quantum systems, e.g., hyperfine atomic states. As another example, in a superconducting quantum computer the qubits may be realized via superconducting qubits or semi-conducting qubits, e.g., superconducting transmon states. As another example, in a NMR quantum computer the qubits may be realized via nuclear spin states.
In some implementations a quantum computation can proceed by initializing the qubits in a selected initial state and applying a sequence of unitary operators on the qubits. Applying a unitary operator to a quantum state can include applying a corresponding sequence of quantum logic gates to the qubits. Example quantum logic gates include single-qubit gates, e.g., Pauli-X, Pauli-Y, Pauli-Z (also referred to as X, Y, Z), Hadamard and S gates, two-qubit gates, e.g., controlled-X, controlled-Y, controlled-Z (also referred to as CX, CY, CZ), and gates involving three or more qubits, e.g., Toffoli gates. The quantum logic gates can be implemented by applying control signals 210 generated by the control and measurement system 204 to the qubits and to the couplers.
For example, in some implementations the qubits in the qubit assembly 252 can be frequency tuneable. In these examples, each qubit can have associated operating frequencies that can be adjusted through application of voltage pulses via one or more drive-lines coupled to the qubit. Example operating frequencies include qubit idling frequencies, qubit interaction frequencies, and qubit readout frequencies. Different frequencies correspond to different operations that the qubit can perform. For example, setting the operating frequency to a corresponding idling frequency may put the qubit into a state where it does not strongly interact with other qubits, and where it may be used to perform single-qubit gates. As another example, in cases where qubits interact via couplers with fixed coupling, qubits can be configured to interact with one another by setting their respective operating frequencies at some gate-dependent frequency detuning from their common interaction frequency. In other cases, e.g., when the qubits interact via tuneable couplers, qubits can be configured to interact with one another by setting the parameters of their respective couplers to enable interactions between the qubits and then by setting the qubit's respective operating frequencies at some gate-dependent frequency detuning from their common interaction frequency. Such interactions may be performed in order to perform multi-qubit gates.
The type of control signals 210 used depends on the physical realizations of the qubits. For example, the control signals may include RF or microwave pulses in an NMR or superconducting quantum computer system, or optical pulses in an atomic quantum computer system.
A quantum computation can be completed by measuring the states of the qubits, e.g., using a quantum observable such as X or Z, using respective control signals 210 . The measurements cause readout signals 212 representing measurement results to be communicated back to the measurement and control system 204 . The readout signals 212 may include RF, microwave, or optical signals depending on the physical scheme for the quantum computing device and/or the qubits. For convenience, the control signals 210 and readout signals 212 shown in FIG. 2 are depicted as addressing only selected elements of the qubit assembly (i.e. the top and bottom rows), but during operation the control signals 210 and readout signals 212 can address each element in the qubit assembly 252 .
The control and measurement system 204 is an example of a classical computer system that can be used to perform various operations on the qubit assembly 252 , as described above, as well as other classical subroutines or computations. The control and measurement system 204 includes one or more classical processors, e.g., classical processor 214 , one or more memories, e.g., memory 216 , and one or more I/O units, e.g., I/O unit 218 , connected by one or more data buses. The control and measurement system 204 can be programmed to send sequences of control signals 210 to the qubit assembly, e.g. to carry out a selected series of quantum gate operations, and to receive sequences of readout signals 212 from the qubit assembly, e.g. as part of performing measurement operations.
The processor 214 is configured to process instructions for execution within the control and measurement system 204 . In some implementations, the processor 214 is a single-threaded processor. In other implementations, the processor 214 is a multi-threaded processor. The processor 214 is capable of processing instructions stored in the memory 216 .
The memory 216 stores information within the control and measurement system 204 . In some implementations, the memory 216 includes a computer-readable medium, a volatile memory unit, and/or a non-volatile memory unit. In some cases, the memory 216 can include storage devices capable of providing mass storage for the system 204 , e.g. a hard disk device, an optical disk device, a storage device that is shared over a network by multiple computing devices (e.g., a cloud storage device), and/or some other large capacity storage device.
The input/output device 218 provides input/output operations for the control and measurement system 204 . The input/output device 218 can include D/A converters, A/D converters, and RF/microwave/optical signal generators, transmitters, and receivers, whereby to send control signals 210 to and receive readout signals 212 from the qubit assembly, as appropriate for the physical scheme for the quantum computer. In some implementations, the input/output device 218 can also include one or more network interface devices, e.g., an Ethernet card, a serial communication device, e.g., an RS-232 port, and/or a wireless interface device, e.g., an 802.11 card. In some implementations, the input/output device 218 can include driver devices configured to receive input data and send output data to other external devices, e.g., keyboard, printer and display devices.
Although an example control and measurement system 204 has been depicted in FIG. 2 , implementations of the subject matter and the functional operations described in this specification can be implemented in other types of digital electronic circuitry, or in computer software, firmware, or hardware, including the structures disclosed in this specification and their structural equivalents, or in combinations of one or more of them.
The example system 200 includes an example classical processor 250 . The classical processor 250 can be used to perform classical computation operations described in this specification according to some implementations, e.g., the classical machine learning methods described herein.
FIG. 3 shows a block diagram of example system 200 of FIG. 2 performing classification and regression tasks using a projected quantum kernel method. Stages (A)-(E) represent a training phase and correspond to steps 402 - 406 of example process 400 as described below with reference to FIG. 4 . During stage (A) of the example process, the quantum computing device 202 obtains a training dataset of data points. In some implementations the data points can be quantum data points, e.g., quantum states. In other implementations the data points can be classical data points. In these implementations, during stage (B), the quantum computing device embeds the classical data points in respective quantum states. Stages (A) and (B) are described in more detail below with reference to step 402 of example process 400 . In some implementations, the training dataset of data points may be received from a classical computer, such as classical processor 250 . In other implementations, the training dataset of data points may be received from a quantum computing device, such as quantum computing device 202 .
During stage (C), the quantum computing device 102 computes a kernel matrix using a kernel function that is based on reduced density matrices for the obtained quantum data points/states. Stage (C) is described in more detail below with reference to step 404 of example process 400 .
During stage (D) the quantum computing device 202 sends the computed kernel matrix to the classical processor 250 . During stage (E), the classical processor receives the kernel matrix and uses the kernel matrix to train a machine learning model.
Stages (F)-(K) represent a verification or inference phase and correspond to steps 408 - 412 of example process 400 . During stage (F) the quantum computing device 202 obtains a validation dataset of data points. In some implementations the data points can be quantum data points, e.g., quantum states. In other implementations the data points can be classical data points. In these implementations, during stage (G), the quantum computing device embeds the classical data points in respective quantum states.
During stage (H) the quantum computing device updates the kernel matrix by computing new rows and columns corresponding to the data points in the validation dataset. Stage (H) is described in more detail below with reference to step 404 and 410 of example process 400 .
During stage (I) the quantum computing device 202 sends the updated kernel matrix to the classical processor 250 . During stage (J), the classical processor receives the updated kernel matrix and processes the updated kernel matrix using the trained machine learning model. During stage (K) the classical processor 250 outputs predictions corresponding to the data points in the validation dataset.
Programming the Hardware
FIG. 4 is a flowchart of an example process 400 for generating and updating a kernel matrix. For convenience, the process 400 will be described as being performed by a system of one or more classical and quantum computing devices located in one or more locations. For example, the quantum computing device 100 of FIG. 1 , appropriately programmed in accordance with this specification, can perform the process 400 .
The quantum computing device obtains a training dataset of quantum data points (step 402 ). The data points can be unlabeled or assigned an associated categorical label or numerical value.
In some implementations the quantum computing device can receive the training dataset as a quantum data input. For example, the quantum computing device can receive a set of quantum states |x i
or access the set of quantum states from a quantum memory included in the quantum computing device. Each quantum state |x i
in the training dataset can be a respective state of an N-qubit quantum system. Each quantum state Ix) can represent a respective classical data point, e.g., an image as described below.
In other implementations the quantum computing device can receive a training dataset of classical data points {x i } and generate a respective training dataset of quantum data points by embedding each classical data point x i into a respective quantum state |x i
. To embed a classical data point x i into an N-qubit quantum state |x i
, the quantum computing device can apply an encoding circuit U enc (x i ) to a reference quantum state of N-qubits, e.g., the state |00 . . . 0 <a href="https://patentimages.storage.googleapis.com/e0/b9/b7/00a4638daedf21/US12321838-20250603-P00001.p
CLAIMS
Claims ( 24 )
What is claimed is:
1. A computer-implemented method comprising:
obtaining, by a quantum computing device, a training dataset of quantum data points;
computing, by the quantum computing device, a kernel matrix that represents similarities among the quantum data points included in the training dataset, wherein the quantum data points are encoded as quantum states of qubits by the quantum computing device, comprising computing, for each pair of quantum data points in the training dataset, a corresponding value of a kernel function, wherein the kernel function is based on reduced density matrices for the quantum data points, wherein the kernel function takes a first and a second quantum data points as input and computes a similarity between the first and the second quantum data points as an element in the kernel matrix, and wherein the reduced density matrices are computed for quantum states of a subset of qubits; and
providing, by the quantum computing device, the kernel matrix to a classical processor.
2. The method of claim 1 , further comprising:
receiving, from the quantum computing device and by the classical processor, the kernel matrix; and
performing, by the classical processor, a training algorithm using the kernel matrix to construct a machine learning model.
3. The method of claim 1 , further comprising:
obtaining, by the quantum computing device, a validation dataset of quantum data points;
computing, by the quantum computing device, new elements of the kernel matrix, wherein the new elements comprise entries representing similarities between the quantum data points in the validation dataset and the quantum data points in the training dataset, wherein the quantum data points in the validation dataset are encoded as quantum states of qubits by the quantum computing device and computing the new elements comprises computing, for each pair of quantum data points in the training dataset and the validation dataset, corresponding values of the kernel function; and
providing, by the quantum computing device, the new elements of the kernel matrix to the classical processor.
4. The method of claim 3 , further comprising processing, by the classical processor, the new elements of the kernel matrix to output predictions for each quantum data point in the validation dataset.
5. The method of claim 1 , wherein the kernel function is based on single-body reduced density matrices for the quantum data points in the training dataset.
6. The method of claim 5 , wherein the kernel function comprises a linear kernel function.
7. The method of claim 6 , wherein the linear kernel function i) takes a first quantum data point and second quantum data point as input, ii) produces a numerical output, and iii) includes a sum of terms, wherein the sum runs over each of N-qubits for N>1 and the summand corresponds to a respective qubit and is equal to the trace of a product of a) a reduced density matrix for the first quantum data point on a subsystem corresponding to the respective qubit b) a reduced density matrix for the second quantum data point on a subsystem corresponding to the respective qubit.
8. The method of claim 6 , wherein the linear kernel function is given by
Q
â¡
(
x
i
â¢
x
j
)
=
â
l
â¢
â¢
Tr
â¡
[
Tr
m
â
l
â¡
[
Ï
â¡
(
x
i
)
]
]
â¡
[
Tr
n
â
l
â¡
[
Ï
â¢
(
x
j
)
]
]
where x i , x j represent a first and second quantum data point, l represents an index that runs from 1 to the number of qubits N and labels each qubit, Ï(x i )=|x i x
x i | and Tr mâ k [Ï(x i )] represents a 1-reduced density matrix (RDM) on qubit k.
9. The method of claim 6 , wherein computing a value of the kernel function for a pair of quantum data points in the training dataset, the pair comprising a first N-qubit quantum state wirth N>1 and a second N-qubit quantum state with N>1, comprises:
repeatedly and for each qubit index:
computing a 1-reduced density matrix (RDM) for the first N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the first N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a first reduced quantum state of the quantum system;
computing a 1-RDM for the second N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the second N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a second reduced quantum state of the quantum system;
determining a trace of the product of the first reduced quantum state and the second reduced quantum state; and
summing averages of the determined traces for each qubit index.
10. The method of claim 5 , wherein the kernel function comprises a squared exponential kernel function.
11. The method of claim 10 , wherein the squared exponential kernel function i) takes a first quantum data point and second quantum data point as input, ii) produces a numerical output, and iii) includes an exponential function of a sum of terms, wherein the sum runs over each of N-qubits for N>1 and the summand corresponds to a respective qubit and is equal to norm of a) a reduced density matrix for the first quantum data point on a subsystem corresponding to the respective qubit minus b) a reduced density matrix for the second quantum data point on a subsystem corresponding to the respective qubit.
12. The method of claim 10 , wherein the squared exponential kernel function is given by
Q
1
E
â¡
(
x
i
â¢
x
j
)
=
exp
(
-
γ
â¢
â
l
â¢
ï
Tr
m
â
l
â¡
[
Ï
â¡
(
x
i
)
]
-
Tr
n
â
l
â¡
[
Ï
â¢
(
x
j
)
]
ï
2
)
where x i , x j represent a first and second quantum data point, I represents an index that runs from 1 to the number of qubits N and labels each qubit, Ï(x i )=|x i
x i | and Tr mâ k [Ï(x i )] represents a 1-RDM on qubit k.
13. The method of claim 10 , wherein computing a value of the kernel function for a pair of quantum data points in the training dataset, the pair comprising a first N-qubit quantum state with N>1 and a second N-qubit quantum state with N>1, comprises, repeatedly and for each qubit index:
computing a 1-reduced density matrix (RDM) for the first N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the first N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a first reduced quantum state of the quantum system;
computing a 1-RDM for the second N-qubit quantum state on a subsystem corresponding to qubit l, comprising obtaining a copy of an N-qubit quantum system in the second N-qubit quantum state and measuring each qubit in the quantum system except the l-th qubit to obtain a second reduced quantum state of the quantum system;
subtracting the second reduced quantum state from the first reduced quantum state to obtain a third reduced quantum state and determining a norm of the third reduced quantum state; and
summing averages of the determined norms for each qubit index and computing an exponent of the summed averages.
14. The method of claim 1 , wherein the kernel function is based on k-body RDMs for the quantum data points, wherein k is less than a predetermined value.
15. The method of claim 14 , wherein the kernel function comprises a linear kernel function.
16. The method of claim 15 , wherein the linear kernel function i) takes a first quantum data point and second quantum data point as input, ii) produces a numerical output, and iii) comprises a sum of terms, wherein the sum runs over each subset of k-qubits taken from the N qubits and each summand corresponds to a respective subset and is equal to the trace of a product of a) a reduced density matrix for the first quantum data point on a subsystem corresponding to the respective subset of k qubits and b) a reduced density matrix for the second quantum data point on a subsystem corresponding to the respective subset of k qubits.
17. The method of claim 15 , wherein the linear kernel function is given by
Q
L
k
â¡
(
x
i
â¢
x
j
)
=
â
K
â
S
k
â¡
(
n
)
â¢
â¢
Tr
â¡
[
Tr
m
â
K
â¡
[
Ï
â¢
(
x
i
)
]
]
â¡
[
Tr
n
â
K
â¡
[
Ï
â¢
(
x
j
)
]
]
where S k (n) represents a set of subsets of k qubits, Ï(x i )=|x i
x i | and Tr mâ K [Ï(x i )] represents a k-RDM.
18. The method claim 15 , wherein computing a value of the kernel function for a pair of quantum data points in the training dataset, the pair comprising a first N-qubit quantum state and a second N-qubit quantum state, comprises:
repeatedly and for each set of k-qubits:
computing a k-RDM for the first N-qubit quantum state on a subsystem corresponding to qubits in the set, comprising obtaining a copy of an N-qubit quantum system in the first N-qubit quantum state and measuring each qubit in the quantum system except for the qubits included in the set to obtain a first reduced quantum state of the quantum system;
computing a k-RDM for the second N-qubit quantum state on a subsystem corresponding to qubits in the set, comprising obtaining a copy of an N-qubit quantum system in the second N-qubit quantum state and measuring each qubit in the quantum system except for the qubits included in the set to obtain a second reduced quantum state of the quantum system;
determining a trace of the product of the first reduced quantum state and the second reduced quantum state; and
summing averages of the determined for each set of k qubits.
19. The method of claim 14 , wherein the kernel function comprises an exponential kernel function.
20. The method of claim 19 , wherein the exponential kernel function is given by
Q
s
â¡
(
x
i
â¢
x
j
)
=
â
k
=
0
â
â¢
â¢
γ
k
k
!
â¢
n
k
â¢
Q
l
k
â¡
(
x
i
â¢
x
j
)
=
ð¼
â¢
â¢
exp
â¡
(
γ
N
â¢
â
h
=
1
n
â¢
(
9
â¢
δ
s
h
i
â¢
s
h
j
â¢
δ
b
h
i
â¢
b
h
j
-
4
)
)
where the expected value E is taken over n s samples from randomly chosen Pauli frames that are measured on a first and second system i and j,
δ
s
h
i
â¢
s
h
j
represents a first indicator function for an agreement between a random Pauli measurement result performed independently on the first system i and the second system j and
δ
b
h
i
â¢
b
h
j
represents a second indicator function for a measurement basis agreement.
21. The method of claim 20 , wherein computing a value of the kernel function for a pair of quantum data points in the training dataset, the pair comprising a first N-qubit quantum state and a second N-qubit quantum state, comprises repeatedly:
obtaining a first measurement result, comprising measuring each qubit in the first system in a random Pauli basis to obtain values s h i and b h i for the h-th qubit, wherein s h i is either 1 or â1 and by is a random basis X, Y, or Z;
obtaining a second measurement result, comprising measuring each qubit in the second system in a random Pauli basis to obtain values s h j and b h j for the h-th qubit, wherein s h j is either 1 or â1 and b h j is a random basis X, Y, or Z;
comparing, for the h-th qubit in the N qubit system, the first measurement result and second measurement result to determine a value of the first indicator function;
determining, for the h-th qubit in the N qubit system, a value of the second indicator function; and
multiplying, summing and averaging the determined values of the first indicator function and the second indicator function.
22. The method of claim 1 , wherein the quantum data points comprise N-qubit quantum states with N>1.
23. The method of claim 1 , wherein obtaining the training dataset of quantum data points comprises:
receiving a training dataset of classical data points; and
generating the training dataset of quantum data points, comprising embedding each classical data point in a respective quantum state by applying a respective encoding circuit to a reference quantum state.
24. An apparatus comprising:
one or more classical processors; and
one or more quantum computing devices in data communication with the one or more classical processors, wherein the quantum computing hardware comprises:
one or more qubit registers, each qubit register comprising one or more qubits, and
a plurality of control devices configured to operate the one or more qubit registers;
wherein the apparatus is configured to perform operations comprising:
obtaining, by a quantum computing device, a training dataset of quantum data points;
computing, by the quantum computing device, a kernel matrix that represents similarities among the quantum data points included in the training dataset, wherein the quantum data points are encoded as quantum states of qubits by the quantum computing device, comprising computing, for each pair of quantum data points in the training dataset, a corresponding value of a kernel function, wherein the kernel function is based on reduced density matrices for the quantum data points, wherein the kernel function takes a first and a second quantum data points as input and computes a similarity between the first and the second quantum data points as an element in the kernel matrix, and wherein the reduced density matrices are computed for quantum states of a subset of qubits; and
providing, by the quantum computing device, the kernel matrix to a classical processor.
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