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Efficient State Read-out for Quantum Machine Learning Algorithms

Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao

Published Apr 14, 2020• 2 citations
OpenAlex

Abstract

Many quantum machine learning (QML) algorithms that claim speed-up over their classical counterparts only generate quantum states as solutions instead of their final classical description. The additional step to decode quantum states into classical vectors normally will destroy the quantum advantage in most scenarios because all existing tomographic methods require runtime that is polynomial with respect to the state dimension. In this Letter, we present an efficient readout protocol that yields the classical vector form of the generated state, so it will achieve the end-to-end advantage for those quantum algorithms. Our protocol suits the case that the output state lies in the row space of the input matrix, of rank $r$, that is stored in the quantum random access memory. The quantum resources for decoding the state in $\ell_2$-norm with $\epsilon$ error require $\text{poly}(r,1/\epsilon)$ copies of the output state and $\text{poly}(r, \kappa^r,1/\epsilon)$ queries to the input oracles, where $\kappa$ is the condition number of the input matrix. With our read-out protocol, we completely characterise the end-to-end resources for quantum linear equation solvers and quantum singular value decomposition. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure, which we believe, will be of independent interest.

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Many quantum algorithms that claim speedup over their classical counterparts only generates quantum states as solutions instead of their final classical description. The additional step to decode quantum states into classical vectors normally will destroy the quantum advantage in most scenarios because all existing tomographic methods require runtime that is polynomial with respect to the state dimension. In this work, we present an efficient readout protocol that yields the classical vector form of the generated state, so it will achieve the end-to-end advantage for those quantum algorithms. Our protocol suits the case in which the output state lies in the row space of the input matrix, of rank $r$, that is stored in the quantum random access memory. The quantum resources for decoding the state in ${\ensuremath{\ell}}^{2}$ norm with $\ensuremath{\epsilon}$ error require $\mathrm{poly}(r,1/\ensuremath{\epsilon})$ copies of the output state and $\mathrm{poly}(r,{\ensuremath{\kappa}}^{r},1/\ensuremath{\epsilon})$ queries to the input oracles, where $\ensuremath{\kappa}$ is the condition number of the input matrix. With our readout protocol, we completely characterize the end-to-end resources for quantum linear equation solvers and quantum singular value decomposition. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure, which we believe will be of independent interest.

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Quantum Algorithms for Linear Algebra and Machine Learning.

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Most quantum algorithms offering speedups over classical algorithms are based on the three techniques of phase estimation, amplitude estimation and Hamiltonian simulation. In spite of the linear algebraic nature of the postulates of quantum mechanics, until recent work by Lloyd and coauthors cite{LMR13, LMR13a, LMR13b} no quantum algorithms achieving speedups for linear algebra or machine learning had been proposed. A quantum machine learning algorithm must address three issues: encoding of classical data into a succinct quantum representation, processing the quantum representation and extraction of classically useful information from the processed quantum state. In this dissertation, we make progress on all three aspects of the quantum machine learning problem and obtain quantum algorithms for low rank approximation and regularized least squares. The oracle $QRAM$, the standard model studied in quantum query complexity, requires time $O(sqrt{n})$ to encode vectors $v in R^{n}$ into quantum states. We propose simple hardware augmentations to the oracle $QRAM$, that enable vectors $v in R^{n}$ to be encoded in time $O(log n)$, with pre-processing. The augmented $QRAM$ incurs minimal hardware overheads, the pre-processing can be parallelized and is a flexible model that allows storage of multiple vectors and matrices. It provides a framework for designing quantum algorithms for linear algebra and machine learning. Using the augmented $QRAM$ for vector state preparation, we present two different algorithms for singular value estimation where given singular vector $ket{v}$ for $A in R^{mtimes n}$, the singular value $sigma_{i}$ is estimated within additive error $epsilon norm{A}_{F}$. The first algorithm requires time $wt{1/epsilon^{3}}$ and uses the approach for simulating $e^{-i rho}$ in cite{LMR13}. However, the analysis cite{LMR13} does not establish the coherence of outputs, we provide a qualitatively different analysis that uses the quantum Zeno effect to establish coherence and reveals the probabilistic nature of the simulation technique. The second algorithm has a running time $wt{1/epsilon}$ and uses Jordan's lemma from linear algebra and the augmented $QRAM$ to implement reflections. We use quantum singular value estimation to obtain algorithms for low rank approximation by column selection, the algorithms are based on importance sampling from the leverage score distribution. We obtain quadratic speedups for a large class of linear algebra algorithms that rely on importance sampling from the leverage score distribution including approximate least squares and $CX$ and $CUR$ decompositions. Classical algorithms for these problems require time $O(mn log n + poly(1/epsilon))$, the quantum algorithms have running time $O(sqrt{m}poly(1/epsilon, k, Delta))$ where $k, Delta$ are the rank and spectral gap. The running time of the quantum $CX$ decomposition algorithm does not depend on $m$, it is polynomial in problem parameters. We also provide quantum algorithms for $ell_{2}$ regularized regression problems, the quantum ridge regression algorithm requires time $wt{1/mu^{2} delta}$ to output a quantum state that is $delta$ close to the solution, where $mu$ is the regularization parameter.

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Along with the development of quantum technology, finding useful applications of quantum computers has been a central pursuit. Despite various quantum algorithms have been developed, many of them often require strong input assumptions, which is hardware demanding. In particular, recent advances on dequantization have revealed that the quantum advantage is more of a mere artifact of strong input assumptions. In this work, we propose a variant of these algorithms, leveraging both classical and quantum resources. Provided the classical knowledge (the entries) of the matrix/vector of interest, a classical procedure is used to pre-process this information. Then they are fed into a quantum circuit which is shown to be a block encoding of the matrix of interest. From this block-encoding, we show how to use it to tackle a wide range of problems, including principal component analysis, linear equation solving, Hamiltonian simulation, preparing ground state, and data fitting. We also analyze our protocol, showing that both the classical and quantum procedure can achieve logarithmic complexity in the input dimension, thus implying its potential for near term realization. We then discuss several implications and corollaries of our result. First,, our results suggest there are certain matrices/Hamiltonians where our method can provide exponential improvement compared to the existing ones with respect to the sparsity. Regarding dense linear systems, our method achieves exponential speed-up with respect to the inverse of error tolerance, compared to the best previously known quantum algorithm for dense systems. Last, and most importantly, regarding quantum data fitting, we show how the output of our quantum algorithms can be leveraged to predict unseen data. Thus, it provides an end-to-end application, which has been an open aspect of the previous quantum data fitting algorithm.

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Most quantum algorithms offering speedups over classical algorithms are based on the three techniques of phase estimation, amplitude estimation and Hamiltonian simulation. In spite of the linear algebraic nature of the postulates of quantum mechanics, until recent work by Lloyd and coauthors cite{LMR13, LMR13a, LMR13b} no quantum algorithms achieving speedups for linear algebra or machine learning had been proposed. A quantum machine learning algorithm must address three issues: encoding of classical data into a succinct quantum representation, processing the quantum representation and extraction of classically useful information from the processed quantum state. In this dissertation, we make progress on all three aspects of the quantum machine learning problem and obtain quantum algorithms for low rank approximation and regularized least squares. The oracle $QRAM$, the standard model studied in quantum query complexity, requires time $O(sqrt{n})$ to encode vectors $v in R^{n}$ into quantum states. We propose simple hardware augmentations to the oracle $QRAM$, that enable vectors $v in R^{n}$ to be encoded in time $O(log n)$, with pre-processing. The augmented $QRAM$ incurs minimal hardware overheads, the pre-processing can be parallelized and is a flexible model that allows storage of multiple vectors and matrices. It provides a framework for designing quantum algorithms for linear algebra and machine learning. Using the augmented $QRAM$ for vector state preparation, we present two different algorithms for singular value estimation where given singular vector $ket{v}$ for $A in R^{mtimes n}$, the singular value $sigma_{i}$ is estimated within additive error $epsilon norm{A}_{F}$. The first algorithm requires time $wt{1/epsilon^{3}}$ and uses the approach for simulating $e^{-i rho}$ in cite{LMR13}. However, the analysis cite{LMR13} does not establish the coherence of outputs, we provide a qualitatively different analysis that uses the quantum Zeno effect to establish coherence and reveals the probabilistic nature of the simulation technique. The second algorithm has a running time $wt{1/epsilon}$ and uses Jordan's lemma from linear algebra and the augmented $QRAM$ to implement reflections. We use quantum singular value estimation to obtain algorithms for low rank approximation by column selection, the algorithms are based on importance sampling from the leverage score distribution. We obtain quadratic speedups for a large class of linear algebra algorithms that rely on importance sampling from the leverage score distribution including approximate least squares and $CX$ and $CUR$ decompositions. Classical algorithms for these problems require time $O(mn log n + poly(1/epsilon))$, the quantum algorithms have running time $O(sqrt{m}poly(1/epsilon, k, Delta))$ where $k, Delta$ are the rank and spectral gap. The running time of the quantum $CX$ decomposition algorithm does not depend on $m$, it is polynomial in problem parameters. We also provide quantum algorithms for $ell_{2}$ regularized regression problems, the quantum ridge regression algorithm requires time $wt{1/mu^{2} delta}$ to output a quantum state that is $delta$ close to the solution, where $mu$ is the regularization parameter.

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Many quantum machine learning (QML) algorithms that claim speed-up over their classical counterparts only generate quantum states as solutions instead of their final classical description. The additional step to decode quantum states into classical vectors normally will destroy the quantum advantage in most scenarios because all existing tomographic methods require runtime that is polynomial with respect to the state dimension. In this Letter, we present an efficient readout protocol that yields the classical vector form of the generated state, so it will achieve the end-to-end advantage for those quantum algorithms. Our protocol suits the case that the output state lies in the row space of the input matrix, of rank rrr, that is stored in the quantum random access memory. The quantum resources for decoding the state in ℓ2\ell_2ℓ2​-norm with ϵ\epsilonϵ error require poly(r,1/ϵ)\text{poly}(r,1/\epsilon)poly(r,1/ϵ) copies of the output state and poly(r,κr,1/ϵ)\text{poly}(r, \kappa^r,1/\epsilon)poly(r,κr,1/ϵ) queries to the input oracles, where κ\kappaκ is the condition number of the input matrix. With our read-out protocol, we completely characterise the end-to-end resources for quantum linear equation solvers and quantum singular value decomposition. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure, which we believe, will be of independent interest.
Many quantum algorithms that claim speedup over their classical counterparts only generates quantum states as solutions instead of their final classical description. The additional step to decode quantum states into classical vectors normally will destroy the quantum advantage in most scenarios because all existing tomographic methods require runtime that is polynomial with respect to the state dimension. In this work, we present an efficient readout protocol that yields the classical vector form of the generated state, so it will achieve the end-to-end advantage for those quantum algorithms. Our protocol suits the case in which the output state lies in the row space of the input matrix, of rank rrr, that is stored in the quantum random access memory. The quantum resources for decoding the state in ${\ensuremath{\ell}}^{2}$ norm with $\ensuremath{\epsilon}$ error require $\mathrm{poly}(r,1/\ensuremath{\epsilon})$ copies of the output state and $\mathrm{poly}(r,{\ensuremath{\kappa}}^{r},1/\ensuremath{\epsilon})$ queries to the input oracles, where $\ensuremath{\kappa}$ is the condition number of the input matrix. With our readout protocol, we completely characterize the end-to-end resources for quantum linear equation solvers and quantum singular value decomposition. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure, which we believe will be of independent interest.
Most quantum algorithms offering speedups over classical algorithms are based on the three techniques of phase estimation, amplitude estimation and Hamiltonian simulation. In spite of the linear algebraic nature of the postulates of quantum mechanics, until recent work by Lloyd and coauthors cite{LMR13, LMR13a, LMR13b} no quantum algorithms achieving speedups for linear algebra or machine learning had been proposed. A quantum machine learning algorithm must address three issues: encoding of classical data into a succinct quantum representation, processing the quantum representation and extraction of classically useful information from the processed quantum state. In this dissertation, we make progress on all three aspects of the quantum machine learning problem and obtain quantum algorithms for low rank approximation and regularized least squares. The oracle QRAMQRAMQRAM, the standard model studied in quantum query complexity, requires time O(sqrtn)O(sqrt{n})O(sqrtn) to encode vectors vinRnv in R^{n}vinRn into quantum states. We propose simple hardware augmentations to the oracle QRAMQRAMQRAM, that enable vectors vinRnv in R^{n}vinRn to be encoded in time O(logn)O(log n)O(logn), with pre-processing. The augmented QRAMQRAMQRAM incurs minimal hardware overheads, the pre-processing can be parallelized and is a flexible model that allows storage of multiple vectors and matrices. It provides a framework for designing quantum algorithms for linear algebra and machine learning. Using the augmented QRAMQRAMQRAM for vector state preparation, we present two different algorithms for singular value estimation where given singular vector ketvket{v}ketv for AinRmtimesnA in R^{mtimes n}AinRmtimesn, the singular value sigmaisigma_{i}sigmai​ is estimated within additive error epsilonnormAFepsilon norm{A}_{F}epsilonnormAF​. The first algorithm requires time wt1/epsilon3wt{1/epsilon^{3}}wt1/epsilon3 and uses the approach for simulating e−irhoe^{-i rho}e−irho in cite{LMR13}. However, the analysis cite{LMR13} does not establish the coherence of outputs, we provide a qualitatively different analysis that uses the quantum Zeno effect to establish coherence and reveals the probabilistic nature of the simulation technique. The second algorithm has a running time wt1/epsilonwt{1/epsilon}wt1/epsilon and uses Jordan's lemma from linear algebra and the augmented QRAMQRAMQRAM to implement reflections. We use quantum singular value estimation to obtain algorithms for low rank approximation by column selection, the algorithms are based on importance sampling from the leverage score distribution. We obtain quadratic speedups for a large class of linear algebra algorithms that rely on importance sampling from the leverage score distribution including approximate least squares and CXCXCX and CURCURCUR decompositions. Classical algorithms for these problems require time O(mnlogn+poly(1/epsilon))O(mn log n + poly(1/epsilon))O(mnlogn+poly(1/epsilon)), the quantum algorithms have running time O(sqrtmpoly(1/epsilon,k,Delta))O(sqrt{m}poly(1/epsilon, k, Delta))O(sqrtmpoly(1/epsilon,k,Delta)) where k,Deltak, Deltak,Delta are the rank and spectral gap. The running time of the quantum CXCXCX decomposition algorithm does not depend on mmm, it is polynomial in problem parameters. We also provide quantum algorithms for ell2ell_{2}ell2​ regularized regression problems, the quantum ridge regression algorithm requires time wt1/mu2deltawt{1/mu^{2} delta}wt1/mu2delta to output a quantum state that is deltadeltadelta close to the solution, where mumumu is the regularization parameter.
Along with the development of quantum technology, finding useful applications of quantum computers has been a central pursuit. Despite various quantum algorithms have been developed, many of them often require strong input assumptions, which is hardware demanding. In particular, recent advances on dequantization have revealed that the quantum advantage is more of a mere artifact of strong input assumptions. In this work, we propose a variant of these algorithms, leveraging both classical and quantum resources. Provided the classical knowledge (the entries) of the matrix/vector of interest, a classical procedure is used to pre-process this information. Then they are fed into a quantum circuit which is shown to be a block encoding of the matrix of interest. From this block-encoding, we show how to use it to tackle a wide range of problems, including principal component analysis, linear equation solving, Hamiltonian simulation, preparing ground state, and data fitting. We also analyze our protocol, showing that both the classical and quantum procedure can achieve logarithmic complexity in the input dimension, thus implying its potential for near term realization. We then discuss several implications and corollaries of our result. First,, our results suggest there are certain matrices/Hamiltonians where our method can provide exponential improvement compared to the existing ones with respect to the sparsity. Regarding dense linear systems, our method achieves exponential speed-up with respect to the inverse of error tolerance, compared to the best previously known quantum algorithm for dense systems. Last, and most importantly, regarding quantum data fitting, we show how the output of our quantum algorithms can be leveraged to predict unseen data. Thus, it provides an end-to-end application, which has been an open aspect of the previous quantum data fitting algorithm.
Most quantum algorithms offering speedups over classical algorithms are based on the three techniques of phase estimation, amplitude estimation and Hamiltonian simulation. In spite of the linear algebraic nature of the postulates of quantum mechanics, until recent work by Lloyd and coauthors cite{LMR13, LMR13a, LMR13b} no quantum algorithms achieving speedups for linear algebra or machine learning had been proposed. A quantum machine learning algorithm must address three issues: encoding of classical data into a succinct quantum representation, processing the quantum representation and extraction of classically useful information from the processed quantum state. In this dissertation, we make progress on all three aspects of the quantum machine learning problem and obtain quantum algorithms for low rank approximation and regularized least squares. The oracle QRAMQRAMQRAM, the standard model studied in quantum query complexity, requires time O(sqrtn)O(sqrt{n})O(sqrtn) to encode vectors vinRnv in R^{n}vinRn into quantum states. We propose simple hardware augmentations to the oracle QRAMQRAMQRAM, that enable vectors vinRnv in R^{n}vinRn to be encoded in time O(logn)O(log n)O(logn), with pre-processing. The augmented QRAMQRAMQRAM incurs minimal hardware overheads, the pre-processing can be parallelized and is a flexible model that allows storage of multiple vectors and matrices. It provides a framework for designing quantum algorithms for linear algebra and machine learning. Using the augmented QRAMQRAMQRAM for vector state preparation, we present two different algorithms for singular value estimation where given singular vector ketvket{v}ketv for AinRmtimesnA in R^{mtimes n}AinRmtimesn, the singular value sigmaisigma_{i}sigmai​ is estimated within additive error epsilonnormAFepsilon norm{A}_{F}epsilonnormAF​. The first algorithm requires time wt1/epsilon3wt{1/epsilon^{3}}wt1/epsilon3 and uses the approach for simulating e−irhoe^{-i rho}e−irho in cite{LMR13}. However, the analysis cite{LMR13} does not establish the coherence of outputs, we provide a qualitatively different analysis that uses the quantum Zeno effect to establish coherence and reveals the probabilistic nature of the simulation technique. The second algorithm has a running time wt1/epsilonwt{1/epsilon}wt1/epsilon and uses Jordan's lemma from linear algebra and the augmented QRAMQRAMQRAM to implement reflections. We use quantum singular value estimation to obtain algorithms for low rank approximation by column selection, the algorithms are based on importance sampling from the leverage score distribution. We obtain quadratic speedups for a large class of linear algebra algorithms that rely on importance sampling from the leverage score distribution including approximate least squares and CXCXCX and CURCURCUR decompositions. Classical algorithms for these problems require time O(mnlogn+poly(1/epsilon))O(mn log n + poly(1/epsilon))O(mnlogn+poly(1/epsilon)), the quantum algorithms have running time O(sqrtmpoly(1/epsilon,k,Delta))O(sqrt{m}poly(1/epsilon, k, Delta))O(sqrtmpoly(1/epsilon,k,Delta)) where k,Deltak, Deltak,Delta are the rank and spectral gap. The running time of the quantum CXCXCX decomposition algorithm does not depend on mmm, it is polynomial in problem parameters. We also provide quantum algorithms for ell2ell_{2}ell2​ regularized regression problems, the quantum ridge regression algorithm requires time wt1/mu2deltawt{1/mu^{2} delta}wt1/mu2delta to output a quantum state that is deltadeltadelta close to the solution, where mumumu is the regularization parameter.
Using quantum algorithms to carry out ML tasks is what is known as Quantum Machine Learning (QML) and the methods developed within this field have the potential to outperform their classical counterparts in solving certain learning problems. The development of the field is partly dependent on that of a functional quantum random access memory (QRAM), called for by some of the algorithms devised. Such a device would store data in a superposition and could then be queried when algorithms require it, similarly to its classical counterpart, allowing for efficient data access. Taking an axiomatic approach to QRAM, this thesis provides the main considerations, assumptions and results regarding QRAM and yields a QRAM handbook and comprehensive introduction to the literature pertaining to it.
Solving linear systems of equations is a frequently encountered problem in machine learning and optimization. Given a matrix AAA and a vector b\mathbf{b}b the task is to find the vector x\mathbf{x}x such that Ax=bA\mathbf{x}=\mathbf{b}Ax=b. We describe a quantum algorithm that achieves a sparsity-independent runtime scaling of $\mathcal{O}\mathbf{(}{\ensuremath{\kappa}}^{2}\sqrt{n}\text{polylog}(n)/\ensuremath{\epsilon}\mathbf{)}$ for an $n\ifmmode\times\else\texttimes\fi{}n$ dimensional AAA with bounded spectral norm, where $\ensuremath{\kappa}$ denotes the condition number of AAA, and $\ensuremath{\epsilon}$ is the desired precision parameter. This amounts to a polynomial improvement over known quantum linear system algorithms when applied to dense matrices, and poses a new state of the art for solving dense linear systems on a quantum computer. Furthermore, an exponential improvement is achievable if the rank of AAA is polylogarithmic in the matrix dimension. Our algorithm is built upon a singular value estimation subroutine, which makes use of a memory architecture that allows for efficient preparation of quantum states that correspond to the rows of AAA and the vector of Euclidean norms of the rows of AAA.