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arXiv:1910.02861

Spectral sparsification of matrix inputs as a preprocessing step for quantum algorithms

Steven Herbert, Sathyawageeswar Subramanian

Published Oct 7, 2019• 2 citations
arXivHTMLPDF

Abstract

We study the potential utility of classical techniques of spectral sparsification of graphs as a preprocessing step for digital quantum algorithms, in particular, for Hamiltonian simulation. Our results indicate that spectral sparsification of a graph with $n$ nodes through a sampling method, e.g.\ as in \cite{Spielman2011resistances} using effective resistances, gives, with high probability, a locally computable matrix $\tilde H$ with row sparsity at most $\mathcal{O}(\text{poly}\log n)$. For a symmetric matrix $H$ of size $n$ with $m$ non-zero entries, a one-time classical runtime overhead of $\mathcal{O}(m||H||t\log n/ε)$ expended in spectral sparsification is then found to be useful as a way to obtain a sparse matrix $\tilde H$ that can be used to approximate time evolution $e^{itH}$ under the Hamiltonian $H$ to precision $ε$. Once such a sparsifier is obtained, it could be used with a variety of quantum algorithms in the query model that make crucial use of row sparsity. We focus on the case of efficient quantum algorithms for sparse Hamiltonian simulation, since Hamiltonian simulation underlies, as a key subroutine, several quantum algorithms, including quantum phase estimation and recent ones for linear algebra. Finally, we also give two simple quantum algorithms to estimate the row sparsity of an input matrix, which achieve a query complexity of $\mathcal{O}(n^{3/2})$ as opposed to $\mathcal{O}(n^2)$ that would be required by any classical algorithm for the task.

DOI
10.48550/arxiv.1910.02861

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Annual Symposium on Foundations of Computer Science•Nov 1, 2020•20 citations
We study the potential utility of classical techniques of spectral sparsification of graphs as a preprocessing step for digital quantum algorithms, in particular, for Hamiltonian simulation. Our results indicate that spectral sparsification of a graph with nnn nodes through a sampling method, e.g.\ as in \cite{Spielman2011resistances} using effective resistances, gives, with high probability, a locally computable matrix H~\tilde HH~ with row sparsity at most O(polylog⁡n)\mathcal{O}(\text{poly}\log n)O(polylogn). For a symmetric matrix HHH of size nnn with mmm non-zero entries, a one-time classical runtime overhead of O(m∣∣H∣∣tlog⁡n/ε)\mathcal{O}(m||H||t\log n/ε)O(m∣∣H∣∣tlogn/ε) expended in spectral sparsification is then found to be useful as a way to obtain a sparse matrix H~\tilde HH~ that can be used to approximate time evolution eitHe^{itH}eitH under the Hamiltonian HHH to precision εεε. Once such a sparsifier is obtained, it could be used with a variety of quantum algorithms in the query model that make crucial use of row sparsity. We focus on the case of efficient quantum algorithms for sparse Hamiltonian simulation, since Hamiltonian simulation underlies, as a key subroutine, several quantum algorithms, including quantum phase estimation and recent ones for linear algebra. Finally, we also give two simple quantum algorithms to estimate the row sparsity of an input matrix, which achieve a query complexity of O(n3/2)\mathcal{O}(n^{3/2})O(n3/2) as opposed to O(n2)\mathcal{O}(n^2)O(n2) that would be required by any classical algorithm for the task.
This dissertation presents a study of quantum algorithms for problems that can be posed as matrix function tasks. In Chapter 1 we demonstrate a simple unifying framework for implementing of smooth functions of matrices on a quantum computer. This framework captures a variety of problems that can be solved by evaluating properties of some function of a matrix, and we identify speedups over classical algorithms for some problem classes. The analysis combines ideas from the classical theory of function approximation with the quantum algorithmic primitive of implementing linear combinations of unitary operators. In Chapter 2 we continue this study by looking at the role of sparsity of input matrices in constructing efficient quantum algorithms. We show that classically pre-processing an input matrix by spectral sparsification can be profitable for quantum Hamiltonian simulation algorithms, without compromising the simulation error or complexity. Such preprocessing incurs a one time cost linear in the size of the matrix, but can be exploited to exponentially speed up subsequent subroutines such as inversion. In Chapter 3, we give an application of this theory of matrix functions to the problem of estimating the Renyi entropy of an unknown quantum state. We combine matrix function techniques with mixed state quantum computation in the one-clean qubit model, and are able to bound of the expected runtime of our algorithm in terms of the unknown target quantity. In addition to the theme of analysing the complexity of our algorithms, we also identify instances that are of practical relevance, leading us to some problems of machine learning. In Chapter 4 we investigate kernel based learning methods using random features. We work with the QRAM input model suitable for big data, and show how matrix functions and the quantum Fourier transform can be used to devise a quantum algorithm for sampling random features that are optimised for given input data and choice of kernel. We obtain a potential exponential speedup over the best known classical algorithm even without explicit assumptions of sparsity or low rank. Finally in Chapter 5 we consider the technique of beamsearch decoding used in natural language processing. We work in the query model, and show how quantum search with advice can be used to construct a quantum search decoder that can find the optimal parse (which may for instance be a best translation, or text-to-speech transcript) at least quadratically faster than the best known classical algorithms, and obtain super-quadratic speedups in the expected runtime.
We give quantum algorithms for Hamiltonian simulation, linear differential equations, and linear systems with square-root dependence on sparsity. For an sss-sparse Hermitian matrix with row and column Euclidean norms bounded by ηηη, evolution for time ttt to error εεε uses O(sτ+sτlog⁡(1/ε)+log⁡(1/ε))\mathcal{O}(\sqrt{s}τ+\sqrt{sτ\log(1/ε)}+\log(1/ε))O(s​τ+sτlog(1/ε)​+log(1/ε)) queries, where τ=η∣t∣τ=η|t|τ=η∣t∣. The same bound holds for time-dependent Hamiltonians with coherent time-indexed sparse access and certified time discretisation. Our construction combines an exact entrywise representation with Cayley transducers. A joint lower bound nearly matches the time, sparsity, and precision dependence, ruling out a uniformly additive bound in sτ\sqrt{s}τs​τ and log⁡(1/ε)\log(1/ε)log(1/ε). For dissipative linear differential equations we obtain O(gˉ[s(τ+τ)+1]log⁡(Cgˉ/ε))\mathcal{O}(\bar g[\sqrt{s}(τ+\sqrtτ)+1]\log(C\bar g/ε))O(gˉ​[s​(τ+τ​)+1]log(Cgˉ​/ε)) queries, where τττ is the evolution time multiplied by a norm bound, gˉ\bar ggˉ​ bounds the ratio of combined input and source norms to the final-solution norm, and CCC is an absolute constant. We sharpen this bound by separating Hamiltonian and dissipative components, and treat general constant generators. Lower bounds capture joint parameter dependence for time-dependent equations and precision hardness for constant positive semidefinite dissipation. For linear systems, a Cayley walk and Chebyshev filter give O(κslog⁡(1/ε))\mathcal{O}(κ\sqrt{s}\log(1/ε))O(κs​log(1/ε)) queries when ∥A∥≤1\|A\|\le1∥A∥≤1 and ∥A−1∥≤κ\|A^{-1}\|\leκ∥A−1∥≤κ, and we give an alternative proof of a matching joint lower bound. We also give gate-efficient implementations of time-independent Hamiltonian simulation and the dissipative ODE solver, including Lipschitz time dependence, retaining the query bounds with polylogarithmic overhead in inverse precision given efficient entry decoding, arithmetic, and required source preparations.
Abstract. Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, “spectral sparsification” reduces the number of edges to near-linear in the number of nodes, while approximately preserving the cut and spectral structure of the graph. In this work we demonstrate a polynomial quantum speedup for spectral sparsification and many of its applications. In particular, we give a quantum algorithm that, given a weighted graph with [Formula: see text] nodes and [Formula: see text] edges, outputs a classical description of an [Formula: see text]-spectral sparsifier in sublinear time [Formula: see text]. This contrasts with the optimal classical complexity [Formula: see text]. We also prove that our quantum algorithm is optimal up to polylog-factors. The algorithm builds on a string of existing results on sparsification, graph spanners, quantum algorithms for shortest paths, and efficient constructions for [Formula: see text]-wise independent random strings. Our algorithm implies a quantum speedup for solving Laplacian systems and for approximating a range of cut problems such as min cut and sparsest cut.
We give a quantum algorithm for simulating a ddd-sparse Hermitian Hamiltonian HHH, assuming a known upper bound ΛΛΛ on its maximum column Euclidean norm ∥H∥1→2\|H\|_{1\to2}∥H∥1→2​. For tΛ≥1/2tΛ\ge1/2tΛ≥1/2, simulation with operator-norm error εεε uses O ⁣(tΛd+dlog⁡(2/ε)) O\!\left(tΛ\sqrt d+\sqrt d\log(2/ε)\right) O(tΛd​+d​log(2/ε)) sparse-oracle queries. This removes the subpolynomial overhead in Low's algorithm [STOC 2019], replacing it with an additive logarithmic precision term. For d>1d>1d>1 and tΛ≥log⁡(2/ε)tΛ\ge\log(2/ε)tΛ≥log(2/ε), the bound matches the worst-case lower bound. A known spectral-norm upper bound may also be used in place of ΛΛΛ. The number of 1- and 2-qubit gates is linear in the query scale, up to oracle costs and polynomial overhead in the input bit lengths and logarithmic precision parameters. As applications, we obtain O(κd polylog(κ/ε))O(κ\sqrt d\,\mathrm{polylog}(κ/ε))O(κd​polylog(κ/ε)) queries for solving ddd-sparse quantum linear systems with ∥A∥≤1\|A\|\le1∥A∥≤1 and ∥A−1∥≤κ\|A^{-1}\|\leκ∥A−1∥≤κ, under standard sparse and state-preparation access. We also give a gate-efficient implementation of black-box unitaries with at most ddd nonzero entries per row and column using O(dlog⁡(2/ε))O(\sqrt d\log(2/ε))O(d​log(2/ε)) queries, given sparse access to the unitary and its adjoint. At constant error, the query bound is optimal and yields Θ(N)Θ(\sqrt N)Θ(N​) queries for arbitrary N×NN\times NN×N unitaries, resolving the open question on black-box unitary implementation posed by Berry and Childs [QIC 2012].
We present a quantum algorithm for approximating the real time evolution e−iHt of an arbitrary d-sparse Hamiltonian to error є, given black-box access to the positions and b-bit values of its non-zero matrix entries. The query complexity of our algorithm is O((t√d||H||1 → 2)1+o(1)/єo(1)) with respect to the largest Euclidean row norm ||H||1 → 2, which is shown to be optimal up to subpolynomial factors through a matching lower bound, and it uses a factor O(b) more gates. This provides a polynomial speedup in sparsity for the common case where the spectral norm is known, and generalizes previous approaches which achieve optimal scaling, but with respect to more restrictive parameters. By exploiting knowledge of the spectral norm, our algorithm solves the black-box unitary implementation problem – O(d1/2+o(1)) queries suffice to approximate any d-sparse unitary in the black-box setting, which matches the quantum search lower bound of Ω(√d) queries and improves upon prior art [Berry and Childs, QIP 2010] of Õ(d2/3) queries. Combined with known techniques, we also solve systems of sparse linear equations with condition number κ using O((κ √d)1+o(1)/єo(1)) queries, which is a quadratic improvement in sparsity.
Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, “spectral sparsification” reduces the number of edges to near-linear in the number of nodes, while approximately preserving the cut and spectral structure of the graph. The breakthrough work by Benczúr and Karger (STOC'96) and Spielman and Teng (STOC'04) showed that sparsification can be done optimally in time near-linear in the number of edges of the original graph. In this work we demonstrate a polynomial quantum speedup for spectral sparsification and many of its applications. In particular, we give a quantum algorithm that, given a weighted graph with n nodes and m edges, outputs a classical description of an ε-spectral sparsifier in sublinear time ~O(√{mn/ε). We prove that this is tight up to polylog-factors. The algorithm builds on a string of existing results, most notably sparsification algorithms by Spielman and Srivastava (STOC'08) and Koutis and Xu (TOPC'16), a spanner construction by Thorup and Zwick (STOC'01), a single-source shortest paths quantum algorithm by Dürr et al. (ICALP'04) and an efficient k-wise independent hash construction by Christiani, Pagh and Thorup (STOC'15). Our algorithm implies a quantum speedup for solving Laplacian systems and for approximating a range of cut problems such as min cut and sparsest cut.