Motivated by recent experiments in which specific thermal properties of complex many-body systems were successfully reproduced on a commercially available quantum annealer, we examine the extent to which quantum annealing hardware can reliably sample from the thermal state in a specific basis associated with a target quantum Hamiltonian. We address this question by studying the diagonal thermal properties of the canonical one-dimensional transverse-field Ising model on a D-Wave 2000Q quantum annealing processor. We find that the quantum processor fails to produce the correct expectation values predicted by Quantum Monte Carlo. Comparing to master equation simulations, we find that this discrepancy is best explained by how the measurements at finite transverse fields are enacted on the device. Specifically, measurements at finite transverse field require the system to be quenched from the target Hamiltonian to a Hamiltonian with negligible transverse field, and this quench is too slow. The limitations imposed by such hardware make it an unlikely candidate for thermal sampling, and it remains an open question what thermal expectation values can be robustly estimated in general for arbitrary quantum many-body systems.
{"title":"Testing a Quantum Annealer as a Quantum Thermal Sampler","authors":"Zoe Gonzalez Izquierdo, I. Hen, T. Albash","doi":"10.1145/3464456","DOIUrl":"https://doi.org/10.1145/3464456","url":null,"abstract":"Motivated by recent experiments in which specific thermal properties of complex many-body systems were successfully reproduced on a commercially available quantum annealer, we examine the extent to which quantum annealing hardware can reliably sample from the thermal state in a specific basis associated with a target quantum Hamiltonian. We address this question by studying the diagonal thermal properties of the canonical one-dimensional transverse-field Ising model on a D-Wave 2000Q quantum annealing processor. We find that the quantum processor fails to produce the correct expectation values predicted by Quantum Monte Carlo. Comparing to master equation simulations, we find that this discrepancy is best explained by how the measurements at finite transverse fields are enacted on the device. Specifically, measurements at finite transverse field require the system to be quenched from the target Hamiltonian to a Hamiltonian with negligible transverse field, and this quench is too slow. The limitations imposed by such hardware make it an unlikely candidate for thermal sampling, and it remains an open question what thermal expectation values can be robustly estimated in general for arbitrary quantum many-body systems.","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"18 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114093111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Semantic knowledge graphs are large-scale triple-oriented databases for knowledge representation and reasoning. Implicit knowledge can be inferred by modeling the tensor representations generated from knowledge graphs. However, as the sizes of knowledge graphs continue to grow, classical modeling becomes increasingly computationally resource intensive. This article investigates how to capitalize on quantum resources to accelerate the modeling of knowledge graphs. In particular, we propose the first quantum machine learning algorithm for inference on tensorized data, i.e., on knowledge graphs. Since most tensor problems are NP-hard [18], it is challenging to devise quantum algorithms to support the inference task. We simplify the modeling task by making the plausible assumption that the tensor representation of a knowledge graph can be approximated by its low-rank tensor singular value decomposition, which is verified by our experiments. The proposed sampling-based quantum algorithm achieves speedup with a polylogarithmic runtime in the dimension of knowledge graph tensor.
{"title":"Quantum Machine Learning Algorithm for Knowledge Graphs","authors":"Yunpu Ma, Yuyi Wang, Volker Tresp","doi":"10.1145/3467982","DOIUrl":"https://doi.org/10.1145/3467982","url":null,"abstract":"Semantic knowledge graphs are large-scale triple-oriented databases for knowledge representation and reasoning. Implicit knowledge can be inferred by modeling the tensor representations generated from knowledge graphs. However, as the sizes of knowledge graphs continue to grow, classical modeling becomes increasingly computationally resource intensive. This article investigates how to capitalize on quantum resources to accelerate the modeling of knowledge graphs. In particular, we propose the first quantum machine learning algorithm for inference on tensorized data, i.e., on knowledge graphs. Since most tensor problems are NP-hard [18], it is challenging to devise quantum algorithms to support the inference task. We simplify the modeling task by making the plausible assumption that the tensor representation of a knowledge graph can be approximated by its low-rank tensor singular value decomposition, which is verified by our experiments. The proposed sampling-based quantum algorithm achieves speedup with a polylogarithmic runtime in the dimension of knowledge graph tensor.","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"168 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120871047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Computing eigenvalues of matrices is ubiquitous in numerical linear algebra problems. Currently, fast quantum algorithms for estimating eigenvalues of Hermitian and unitary matrices are known. However, the general case is far from fully understood in the quantum case. Based on a quantum algorithm for solving linear ordinary differential equations, we show how to estimate the eigenvalues of diagonalizable matrices that only have real eigenvalues. The output is a superposition of the eigenpairs, and the overall complexity is polylog in the dimension for sparse matrices. Under an assumption, we extend the algorithm to diagonalizable matrices with complex eigenvalues.
{"title":"Computing Eigenvalues of Diagonalizable Matrices on a Quantum Computer","authors":"Changpeng Shao","doi":"10.1145/3527845","DOIUrl":"https://doi.org/10.1145/3527845","url":null,"abstract":"Computing eigenvalues of matrices is ubiquitous in numerical linear algebra problems. Currently, fast quantum algorithms for estimating eigenvalues of Hermitian and unitary matrices are known. However, the general case is far from fully understood in the quantum case. Based on a quantum algorithm for solving linear ordinary differential equations, we show how to estimate the eigenvalues of diagonalizable matrices that only have real eigenvalues. The output is a superposition of the eigenpairs, and the overall complexity is polylog in the dimension for sparse matrices. Under an assumption, we extend the algorithm to diagonalizable matrices with complex eigenvalues.","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131157581","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose an efficient heuristic for mapping the logical qubits of quantum algorithms to the physical qubits of connectivity-limited devices, adding a minimal number of connectivity-compliant SWAP gates. In particular, given a quantum circuit, we construct an undirected graph with edge weights a function of the two-qubit gates of the quantum circuit. Taking inspiration from spectral graph drawing, we use an eigenvector of the graph Laplacian to place logical qubits at coordinate locations. These placements are then mapped to physical qubits for a given connectivity. We primarily focus on one-dimensional connectivities and sketch how the general principles of our heuristic can be extended for use in more general connectivities.
{"title":"Using Spectral Graph Theory to Map Qubits onto Connectivity-limited Devices","authors":"Joseph X. Lin, Eric R. Anschuetz, A. Harrow","doi":"10.1145/3436752","DOIUrl":"https://doi.org/10.1145/3436752","url":null,"abstract":"We propose an efficient heuristic for mapping the logical qubits of quantum algorithms to the physical qubits of connectivity-limited devices, adding a minimal number of connectivity-compliant SWAP gates. In particular, given a quantum circuit, we construct an undirected graph with edge weights a function of the two-qubit gates of the quantum circuit. Taking inspiration from spectral graph drawing, we use an eigenvector of the graph Laplacian to place logical qubits at coordinate locations. These placements are then mapped to physical qubits for a given connectivity. We primarily focus on one-dimensional connectivities and sketch how the general principles of our heuristic can be extended for use in more general connectivities.","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133704749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Hayato Ushijima-Mwesigwa, Ruslan Shaydulin, C. Negre, S. Mniszewski, Y. Alexeev, Ilya Safro
Emerging quantum processors provide an opportunity to explore new approaches for solving traditional problems in the post Moore’s law supercomputing era. However, the limited number of qubits makes it infeasible to tackle massive real-world datasets directly in the near future, leading to new challenges in utilizing these quantum processors for practical purposes. Hybrid quantum-classical algorithms that leverage both quantum and classical types of devices are considered as one of the main strategies to apply quantum computing to large-scale problems. In this article, we advocate the use of multilevel frameworks for combinatorial optimization as a promising general paradigm for designing hybrid quantum-classical algorithms. To demonstrate this approach, we apply this method to two well-known combinatorial optimization problems, namely, the Graph Partitioning Problem, and the Community Detection Problem. We develop hybrid multilevel solvers with quantum local search on D-Wave’s quantum annealer and IBM’s gate-model based quantum processor. We carry out experiments on graphs that are orders of magnitude larger than the current quantum hardware size, and we observe results comparable to state-of-the-art solvers in terms of quality of the solution. Reproducibility: Our code and data are available at Reference [1].
{"title":"Multilevel Combinatorial Optimization across Quantum Architectures","authors":"Hayato Ushijima-Mwesigwa, Ruslan Shaydulin, C. Negre, S. Mniszewski, Y. Alexeev, Ilya Safro","doi":"10.1145/3425607","DOIUrl":"https://doi.org/10.1145/3425607","url":null,"abstract":"Emerging quantum processors provide an opportunity to explore new approaches for solving traditional problems in the post Moore’s law supercomputing era. However, the limited number of qubits makes it infeasible to tackle massive real-world datasets directly in the near future, leading to new challenges in utilizing these quantum processors for practical purposes. Hybrid quantum-classical algorithms that leverage both quantum and classical types of devices are considered as one of the main strategies to apply quantum computing to large-scale problems. In this article, we advocate the use of multilevel frameworks for combinatorial optimization as a promising general paradigm for designing hybrid quantum-classical algorithms. To demonstrate this approach, we apply this method to two well-known combinatorial optimization problems, namely, the Graph Partitioning Problem, and the Community Detection Problem. We develop hybrid multilevel solvers with quantum local search on D-Wave’s quantum annealer and IBM’s gate-model based quantum processor. We carry out experiments on graphs that are orders of magnitude larger than the current quantum hardware size, and we observe results comparable to state-of-the-art solvers in terms of quality of the solution. Reproducibility: Our code and data are available at Reference [1].","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130275470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can readily solve a quantum linear system problem (QLSP) with O(κ poly(log (κ ε))) runtime, where κ is the condition number, and ε is the target accuracy. This is near optimal with respect to both κ and ε, and is achieved without relying on complicated amplitude amplification procedures that are difficult to implement. Our method is applicable to general non-Hermitian matrices, and the cost as well as the number of qubits can be reduced when restricted to Hermitian matrices, and further to Hermitian positive definite matrices. The success of the time-optimal AQC implies that the quantum approximate optimization algorithm (QAOA) with an optimal control protocol can also achieve the same complexity in terms of the runtime. Numerical results indicate that QAOA can yield the lowest runtime compared to the time-optimal AQC, vanilla AQC, and the recently proposed randomization method.
{"title":"Quantum Linear System Solver Based on Time-optimal Adiabatic Quantum Computing and Quantum Approximate Optimization Algorithm","authors":"Dong An, Lin Lin","doi":"10.1145/3498331","DOIUrl":"https://doi.org/10.1145/3498331","url":null,"abstract":"We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can readily solve a quantum linear system problem (QLSP) with O(κ poly(log (κ ε))) runtime, where κ is the condition number, and ε is the target accuracy. This is near optimal with respect to both κ and ε, and is achieved without relying on complicated amplitude amplification procedures that are difficult to implement. Our method is applicable to general non-Hermitian matrices, and the cost as well as the number of qubits can be reduced when restricted to Hermitian matrices, and further to Hermitian positive definite matrices. The success of the time-optimal AQC implies that the quantum approximate optimization algorithm (QAOA) with an optimal control protocol can also achieve the same complexity in terms of the runtime. Numerical results indicate that QAOA can yield the lowest runtime compared to the time-optimal AQC, vanilla AQC, and the recently proposed randomization method.","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125318022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Raban Iten, Romain Moyard, Tony Metger, David Sutter, Stefan Woerner
Quantum computations are typically performed as a sequence of basic operations, called quantum gates. Different gate sequences, called quantum circuits, can implement the same overall quantum computation. Since every additional quantum gate takes time and introduces noise into the system, it is important to find the smallest possible quantum circuit that implements a given computation, especially for near-term quantum devices that can execute only a limited number of quantum gates before noise renders the computation useless. An important building block for many quantum circuit optimization techniques is pattern matching: given a large and small quantum circuit, we would like to find all maximal matches of the small circuit, called a pattern, in the large circuit, considering pairwise commutation of quantum gates. In this work, we present the first classical algorithm for pattern matching that provably finds all maximal matches and is efficient enough to be practical for circuit sizes typical for near-term devices. We demonstrate numerically1 that combining our algorithm with known pattern-matching-based circuit optimization techniques reduces the gate count of a random quantum circuit by ∼ 30% and can further improve practically relevant quantum circuits that were already optimized with state-of-the-art techniques.
{"title":"Exact and Practical Pattern Matching for Quantum Circuit Optimization","authors":"Raban Iten, Romain Moyard, Tony Metger, David Sutter, Stefan Woerner","doi":"10.1145/3498325","DOIUrl":"https://doi.org/10.1145/3498325","url":null,"abstract":"Quantum computations are typically performed as a sequence of basic operations, called quantum gates. Different gate sequences, called quantum circuits, can implement the same overall quantum computation. Since every additional quantum gate takes time and introduces noise into the system, it is important to find the smallest possible quantum circuit that implements a given computation, especially for near-term quantum devices that can execute only a limited number of quantum gates before noise renders the computation useless. An important building block for many quantum circuit optimization techniques is pattern matching: given a large and small quantum circuit, we would like to find all maximal matches of the small circuit, called a pattern, in the large circuit, considering pairwise commutation of quantum gates. In this work, we present the first classical algorithm for pattern matching that provably finds all maximal matches and is efficient enough to be practical for circuit sizes typical for near-term devices. We demonstrate numerically1 that combining our algorithm with known pattern-matching-based circuit optimization techniques reduces the gate count of a random quantum circuit by ∼ 30% and can further improve practically relevant quantum circuits that were already optimized with state-of-the-art techniques.","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"101 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124648107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shouvanik Chakrabarti, Andrew M. Childs, S. Hung, Tongyang Li, C. Wang, Xiaodi Wu
Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an n-dimensional convex body within multiplicative error ε using Õ(n3 + n2.5/ε) queries to a membership oracle and Õ(n5+n4.5/ε) additional arithmetic operations. For comparison, the best known classical algorithm uses Õ(n3.5+n3/ε2) queries and Õ(n5.5+n5/ε2) additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of “Chebyshev cooling,” where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requires Ω (√ n+1/ε) quantum membership queries, which rules out the possibility of exponential quantum speedup in n and shows optimality of our algorithm in 1/ε up to poly-logarithmic factors.
{"title":"Quantum Algorithm for Estimating Volumes of Convex Bodies","authors":"Shouvanik Chakrabarti, Andrew M. Childs, S. Hung, Tongyang Li, C. Wang, Xiaodi Wu","doi":"10.1145/3588579","DOIUrl":"https://doi.org/10.1145/3588579","url":null,"abstract":"Estimating the volume of a convex body is a central problem in convex geometry and can be viewed as a continuous version of counting. We present a quantum algorithm that estimates the volume of an n-dimensional convex body within multiplicative error ε using Õ(n3 + n2.5/ε) queries to a membership oracle and Õ(n5+n4.5/ε) additional arithmetic operations. For comparison, the best known classical algorithm uses Õ(n3.5+n3/ε2) queries and Õ(n5.5+n5/ε2) additional arithmetic operations. To the best of our knowledge, this is the first quantum speedup for volume estimation. Our algorithm is based on a refined framework for speeding up simulated annealing algorithms that might be of independent interest. This framework applies in the setting of “Chebyshev cooling,” where the solution is expressed as a telescoping product of ratios, each having bounded variance. We develop several novel techniques when implementing our framework, including a theory of continuous-space quantum walks with rigorous bounds on discretization error. To complement our quantum algorithms, we also prove that volume estimation requires Ω (√ n+1/ε) quantum membership queries, which rules out the possibility of exponential quantum speedup in n and shows optimality of our algorithm in 1/ε up to poly-logarithmic factors.","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133488456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pauli channels are ubiquitous in quantum information, both as a dominant noise source in many computing architectures and as a practical model for analyzing error correction and fault tolerance. Here, we prove several results on efficiently learning Pauli channels and more generally the Pauli projection of a quantum channel. We first derive a procedure for learning a Pauli channel on n qubits with high probability to a relative precision ϵ using O(ϵ-2n2n) measurements, which is efficient in the Hilbert space dimension. The estimate is robust to state preparation and measurement errors, which, together with the relative precision, makes it especially appropriate for applications involving characterization of high-accuracy quantum gates. Next, we show that the error rates for an arbitrary set of s Pauli errors can be estimated to a relative precision ϵ using O(ϵ-4log s log s/ϵ) measurements. Finally, we show that when the Pauli channel is given by a Markov field with at most k-local correlations, we can learn an entire n-qubit Pauli channel to relative precision ϵ with only Ok(ϵ-2n2logn) measurements, which is efficient in the number of qubits. These results enable a host of applications beyond just characterizing noise in a large-scale quantum system: they pave the way to tailoring quantum codes, optimizing decoders, and customizing fault tolerance procedures to suit a particular device.
泡利信道在量子信息中无处不在，它既是许多计算体系结构中的主要噪声源，也是分析纠错和容错的实用模型。在这里，我们证明了有效学习泡利信道和量子信道的泡利投影的几个结果。我们首先使用O(ϵ-2n2n)测量，推导出一个在n个量子位上以高概率学习泡利通道的过程，达到相对精度的λ，这在希尔伯特空间维度上是有效的。该估计对状态准备和测量误差具有鲁棒性，加上相对精度，使其特别适用于涉及高精度量子门表征的应用。接下来，我们展示了任意一组s泡利误差的错误率可以使用O(ϵ-4log s log s/ λ)测量来估计到相对精度的λ。最后，我们表明，当泡利通道由最多k个局部相关的马尔可夫场给出时，我们可以仅通过Ok(ϵ-2n2logn)测量就可以将整个n-量子位泡利通道学习到相对精度的λ，这在量子位的数量上是有效的。这些结果使得大量的应用不仅仅是表征大规模量子系统中的噪声:它们为定制量子编码、优化解码器和定制容错程序铺平了道路，以适应特定的设备。
{"title":"Efficient Estimation of Pauli Channels","authors":"S. Flammia, Joel J. Wallman","doi":"10.1145/3408039","DOIUrl":"https://doi.org/10.1145/3408039","url":null,"abstract":"Pauli channels are ubiquitous in quantum information, both as a dominant noise source in many computing architectures and as a practical model for analyzing error correction and fault tolerance. Here, we prove several results on efficiently learning Pauli channels and more generally the Pauli projection of a quantum channel. We first derive a procedure for learning a Pauli channel on n qubits with high probability to a relative precision ϵ using O(ϵ-2n2n) measurements, which is efficient in the Hilbert space dimension. The estimate is robust to state preparation and measurement errors, which, together with the relative precision, makes it especially appropriate for applications involving characterization of high-accuracy quantum gates. Next, we show that the error rates for an arbitrary set of s Pauli errors can be estimated to a relative precision ϵ using O(ϵ-4log s log s/ϵ) measurements. Finally, we show that when the Pauli channel is given by a Markov field with at most k-local correlations, we can learn an entire n-qubit Pauli channel to relative precision ϵ with only Ok(ϵ-2n2logn) measurements, which is efficient in the number of qubits. These results enable a host of applications beyond just characterizing noise in a large-scale quantum system: they pave the way to tailoring quantum codes, optimizing decoders, and customizing fault tolerance procedures to suit a particular device.","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"15 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132287260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Allcock, Chang-Yu Hsieh, Iordanis Kerenidis, Shengyu Zhang
Quantum machine learning has the potential for broad industrial applications, and the development of quantum algorithms for improving the performance of neural networks is of particular interest given the central role they play in machine learning today. We present quantum algorithms for training and evaluating feedforward neural networks based on the canonical classical feedforward and backpropagation algorithms. Our algorithms rely on an efficient quantum subroutine for approximating inner products between vectors in a robust way, and on implicitly storing intermediate values in quantum random access memory for fast retrieval at later stages. The running times of our algorithms can be quadratically faster in the size of the network than their standard classical counterparts since they depend linearly on the number of neurons in the network, and not on the number of connections between neurons. Furthermore, networks trained by our quantum algorithm may have an intrinsic resilience to overfitting, as the algorithm naturally mimics the effects of classical techniques used to regularize networks. Our algorithms can also be used as the basis for new quantum-inspired classical algorithms with the same dependence on the network dimensions as their quantum counterparts but with quadratic overhead in other parameters that makes them relatively impractical.
{"title":"Quantum Algorithms for Feedforward Neural Networks","authors":"J. Allcock, Chang-Yu Hsieh, Iordanis Kerenidis, Shengyu Zhang","doi":"10.1145/3411466","DOIUrl":"https://doi.org/10.1145/3411466","url":null,"abstract":"Quantum machine learning has the potential for broad industrial applications, and the development of quantum algorithms for improving the performance of neural networks is of particular interest given the central role they play in machine learning today. We present quantum algorithms for training and evaluating feedforward neural networks based on the canonical classical feedforward and backpropagation algorithms. Our algorithms rely on an efficient quantum subroutine for approximating inner products between vectors in a robust way, and on implicitly storing intermediate values in quantum random access memory for fast retrieval at later stages. The running times of our algorithms can be quadratically faster in the size of the network than their standard classical counterparts since they depend linearly on the number of neurons in the network, and not on the number of connections between neurons. Furthermore, networks trained by our quantum algorithm may have an intrinsic resilience to overfitting, as the algorithm naturally mimics the effects of classical techniques used to regularize networks. Our algorithms can also be used as the basis for new quantum-inspired classical algorithms with the same dependence on the network dimensions as their quantum counterparts but with quadratic overhead in other parameters that makes them relatively impractical.","PeriodicalId":365166,"journal":{"name":"ACM Transactions on Quantum Computing","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122170060","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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