arXiv subjects

Pawel Wocjan

Publications and source records attributed to Pawel Wocjan.

At least 55 records · Page 3Linked to original sources

arXiv2010

Quantum Algorithm for Preparing Thermal Gibbs States - Detailed Analysis

In a recent work [10], Poulin and one of us presented a quantum algorithm for preparing thermal Gibbs states of interacting quantum systems. This algorithm is based on Grovers's technique for quantum state engineering, and its running time is dominated by the factor D/Z(β), where D and Z(β) denote the dimension of the quantum system and its partition function at inverse temperature β, respectively. We present here a modified algorithm and a more detailed analysis of the errors that arise due to imperfect simulation of Hamiltonian time evolutions and limited performance of phase estimation (finite accuracy and nonzero probability of failure). This modfication together with the tighter analysis allows us to prove a better running time by the effect of these sources of error on the overall complexity. We think that the ideas underlying of our new analysis could also be used to prove a better performance of quantum Metropolis sampling by Temme et al. [12].

physics.comp-ph↗

arXiv2010

Efficient quantum circuits for arbitrary sparse unitaries

Arbitrary exponentially large unitaries cannot be implemented efficiently by quantum circuits. However, we show that quantum circuits can efficiently implement any unitary provided it has at most polynomially many nonzero entries in any row or column, and these entries are efficiently computable. One can formulate a model of computation based on the composition of sparse unitaries which includes the quantum Turing machine model, the quantum circuit model, anyonic models, permutational quantum computation, and discrete time quantum walks as special cases. Thus we obtain a simple unified proof that these models are all contained in BQP. Furthermore our general method for implementing sparse unitaries simplifies several existing quantum algorithms.

quant-ph↗

arXiv2009

Efficient Circuits for Quantum Walks

We present an efficient general method for realizing a quantum walk operator corresponding to an arbitrary sparse classical random walk. Our approach is based on Grover and Rudolph's method for preparing coherent versions of efficiently integrable probability distributions. This method is intended for use in quantum walk algorithms with polynomial speedups, whose complexity is usually measured in terms of how many times we have to apply a step of a quantum walk, compared to the number of necessary classical Markov chain steps. We consider a finer notion of complexity including the number of elementary gates it takes to implement each step of the quantum walk with some desired accuracy. The difference in complexity for various implementation approaches is that our method scales linearly in the sparsity parameter and poly-logarithmically with the inverse of the desired precision. The best previously known general methods either scale quadratically in the sparsity parameter, or polynomially in the inverse precision. Our approach is especially relevant for implementing quantum walks corresponding to classical random walks like those used in the classical algorithms for approximating permanents and sampling from binary contingency tables. In those algorithms, the sparsity parameter grows with the problem size, while maintaining high precision is required.

quant-ph↗

arXiv2009

Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer

We present a quantum algorithm to prepare the thermal Gibbs state of interacting quantum systems. This algorithm sets a universal upper bound D^alpha on the thermalization time of a quantum system, where D is the system's Hilbert space dimension and alpha < 1/2 is proportional to the Helmholtz free energy density of the system. We also derive an algorithm to evaluate the partition function of a quantum system in a time proportional to the system's thermalization time and inversely proportional to the targeted accuracy squared.

quant-ph↗

arXiv2009

Quantum Speed-up for Approximating Partition Functions

We achieve a quantum speed-up of fully polynomial randomized approximation schemes (FPRAS) for estimating partition functions that combine simulated annealing with the Monte-Carlo Markov Chain method and use non-adaptive cooling schedules. The improvement in time complexity is twofold: a quadratic reduction with respect to the spectral gap of the underlying Markov chains and a quadratic reduction with respect to the parameter characterizing the desired accuracy of the estimate output by the FPRAS. Both reductions are intimately related and cannot be achieved separately. First, we use Grover's fixed point search, quantum walks and phase estimation to efficiently prepare approximate coherent encodings of stationary distributions of the Markov chains. The speed-up we obtain in this way is due to the quadratic relation between the spectral and phase gaps of classical and quantum walks. Second, we generalize the method of quantum counting, showing how to estimate expected values of quantum observables. Using this method instead of classical sampling, we obtain the speed-up with respect to accuracy.

quant-ph↗

arXiv2009

Fast Amplification of QMA

Given a verifier circuit for a problem in QMA, we show how to exponentially amplify the gap between its acceptance probabilities in the `yes' and `no' cases, with a method that is quadratically faster than the procedure given by Marriott and Watrous. Our construction is natively quantum, based on the analogy of a product of two reflections and a quantum walk. Second, in some special cases we show how to amplify the acceptance probability for good witnesses to 1, making a step towards the proof that QMA with one-sided error is equal to QMA. Finally, we simplify the filter-state method to search for QMA witnesses by Poulin and Wocjan.

quant-ph↗

arXiv2008

Estimating Jones and HOMFLY polynomials with One Clean Qubit

The Jones and HOMFLY polynomials are link invariants with close connections to quantum computing. It was recently shown that finding a certain approximation to the Jones polynomial of the trace closure of a braid at the fifth root of unity is a complete problem for the one clean qubit complexity class. This is the class of problems solvable in polynomial time on a quantum computer acting on an initial state in which one qubit is pure and the rest are maximally mixed. Here we generalize this result by showing that one clean qubit computers can efficiently approximate the Jones and single-variable HOMFLY polynomials of the trace closure of a braid at any root of unity.

quant-ph↗

arXiv2008

Preparing ground states of quantum many-body systems on a quantum computer

Preparing the ground state of a system of interacting classical particles is an NP-hard problem. Thus, there is in general no better algorithm to solve this problem than exhaustively going through all N configurations of the system to determine the one with lowest energy, requiring a running time proportional to N. A quantum computer, if it could be built, could solve this problem in time sqrt(N). Here, we present a powerful extension of this result to the case of interacting quantum particles, demonstrating that a quantum computer can prepare the ground state of a quantum system as efficiently as it does for classical systems.

quant-ph↗

arXiv2008

Speed-up via Quantum Sampling

The Markov Chain Monte Carlo method is at the heart of efficient approximation schemes for a wide range of problems in combinatorial enumeration and statistical physics. It is therefore very natural and important to determine whether quantum computers can speed-up classical mixing processes based on Markov chains. To this end, we present a new quantum algorithm, making it possible to prepare a quantum sample, i.e., a coherent version of the stationary distribution of a reversible Markov chain. Our algorithm has a significantly better running time than that of a previous algorithm based on adiabatic state generation. We also show that our methods provide a speed-up over a recently proposed method for obtaining ground states of (classical) Hamiltonians.

quant-ph↗

arXiv2008

Efficient Quantum Algorithm for Identifying Hidden Polynomials

We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field F with d elements. The hidden functions of the generalized problem are not restricted to be linear but can also be m-variate polynomial functions of total degree n>=2. The problem of identifying hidden m-variate polynomials of degree less or equal to n for fixed n and m is hard on a classical computer since Omega(sqrt{d}) black-box queries are required to guarantee a constant success probability. In contrast, we present a quantum algorithm that correctly identifies such hidden polynomials for all but a finite number of values of d with constant probability and that has a running time that is only polylogarithmic in d.

quant-ph↗

arXiv2008

Hamiltonian Quantum Cellular Automata in 1D

We construct a simple translationally invariant, nearest-neighbor Hamiltonian on a chain of 10-dimensional qudits that makes it possible to realize universal quantum computing without any external control during the computational process. We only require the ability to prepare an initial computational basis state which encodes both the quantum circuit and its input. The computational process is then carried out by the autonomous Hamiltonian time evolution. After a time polynomially long in the size of the quantum circuit has passed, the result of the computation is obtained with high probability by measuring a few qudits in the computational basis. This result also implies that there cannot exist efficient classical simulation methods for generic translationally invariant nearest-neighbor Hamiltonians on qudit chains, unless quantum computers can be efficiently simulated by classical computers (or, put in complexity theoretic terms, unless BPP=BQP).

quant-ph↗

arXiv2008

A PromiseBQP-complete String Rewriting Problem

We are given three strings s, t, and t' of length L over some fixed finite alphabet and an integer m that is polylogarithmic in L. We have a symmetric relation on substrings of constant length that specifies which substrings are allowed to be replaced with each other. Let Delta(n) denote the difference between the numbers of possibilities to obtain t from s and t' from s after n replacements. The problem is to determine the sign of Delta(m). As promises we have a gap condition and a growth condition. The former states that |Delta(m)| >= epsilon c^m where epsilon is inverse polylogarithmic in L and c>0 is a constant. The latter is given by Delta(n) <= c^n for all n. We show that this problem is PromiseBQP-complete, i.e., it represents the class of problems which can be solved efficiently on a quantum computer.

quant-ph↗

arXiv2007

A single-shot measurement of the energy of product states in a translation invariant spin chain can replace any quantum computation

In measurement-based quantum computation, quantum algorithms are implemented via sequences of measurements. We describe a translationally invariant finite-range interaction on a one-dimensional qudit chain and prove that a single-shot measurement of the energy of an appropriate computational basis state with respect to this Hamiltonian provides the output of any quantum circuit. The required measurement accuracy scales inverse polynomially with the size of the simulated quantum circuit. This shows that the implementation of energy measurements on generic qudit chains is as hard as the realization of quantum computation. Here a ''measurement'' is any procedure that samples from the spectral measure induced by the observable and the state under consideration. As opposed to measurement-based quantum computation, the post-measurement state is irrelevant.

quant-ph↗

arXiv2007

The Jones polynomial: quantum algorithms and applications in quantum complexity theory

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al. that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

quant-ph↗

arXiv2007

Efficient Quantum Algorithm for Hidden Quadratic and Cubic Polynomial Function Graphs

We introduce the Hidden Polynomial Function Graph Problem as a natural generalization of an abelian Hidden Subgroup Problem (HSP) where the subgroups and their cosets correspond to graphs of linear functions over the finite field F_p. For the Hidden Polynomial Function Graph Problem the functions are not restricted to be linear but can also be multivariate polynomial functions of higher degree. For a fixed number of indeterminates and bounded total degree the Hidden Polynomial Function Graph Problem is hard on a classical computer as its black box query complexity is polynomial in p. In contrast, this problem can be reduced to a quantum state identification problem so that the resulting quantum query complexity does not depend on p. For univariate polynomials we construct a von Neumann measurement for distinguishing the states. We relate the success probability and the implementation of this measurement to certain classical problems involving polynomial equations. We present an efficient algorithm for hidden quadratic and cubic function graphs by establishing that the success probability of the measurement is lower bounded by a constant and that it can be implemented efficiently.

quant-ph↗

arXiv2006

BQP-complete Problems Concerning Mixing Properties of Classical Random Walks on Sparse Graphs

We describe two BQP-complete problems concerning properties of sparse graphs having a certain symmetry. The graphs are specified by efficiently computable functions which output the adjacent vertices for each vertex. Let i and j be two given vertices. The first problem consists in estimating the difference between the number of paths of length m from j to j and those which from i to j, where m is polylogarithmic in the number of vertices. The scale of the estimation accuracy is specified by some a priori known upper bound on the growth of these differences with increasing m. The problem remains BQP-hard for regular graphs with degree 4. The second problem is related to continuous-time classical random walks. The walk starts at some vertex j. The promise is that the difference of the probabilities of being at j and at i, respectively, decays with O(exp(-μt)) for some μ>0. The problem is to decide whether this difference is greater than a exp(-μT) or smaller than b exp(-μT) after some time instant T, where T is polylogarithmic and the difference a-b is inverse polylogarithmic in the number of vertices. Since the probabilities differ only by an exponentially small amount, an exponential number of trials would be necessary if one tried to answer this question by running the walk itself. A modification of this problem, asking whether there exists a pair of nodes for which the probability difference is at least a exp(-μT), is QCMA-complete.

quant-ph↗

arXiv2006

Estimating diagonal entries of powers of sparse symmetric matrices is BQP-complete

Let A be a real symmetric matrix of size N such that the number of the non-zero entries in each row is polylogarithmic in N and the positions and the values of these entries are specified by an efficiently computable function. We consider the problem of estimating an arbitrary diagonal entry (A^m)_jj of the matrix A^m up to an error of εb^m, where b is an a priori given upper bound on the norm of A, m and εare polylogarithmic and inverse polylogarithmic in N, respectively. We show that this problem is BQP-complete. It can be solved efficiently on a quantum computer by repeatedly applying measurements of A to the jth basis vector and raising the outcome to the mth power. Conversely, every quantum circuit that solves a problem in BQP can be encoded into a sparse matrix such that some basis vector |j> corresponding to the input induces two different spectral measures depending on whether the input is accepted or not. These measures can be distinguished by estimating the mth statistical moment for some appropriately chosen m, i.e., by the jth diagonal entry of A^m. The problem is still in BQP when generalized to off-diagonal entries and it remains BQP-hard if A has only -1, 0, and 1 as entries.

quant-ph↗

arXiv2006

Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem

Schur duality decomposes many copies of a quantum state into subspaces labeled by partitions, a decomposition with applications throughout quantum information theory. Here we consider applying Schur duality to the problem of distinguishing coset states in the standard approach to the hidden subgroup problem. We observe that simply measuring the partition (a procedure we call weak Schur sampling) provides very little information about the hidden subgroup. Furthermore, we show that under quite general assumptions, even a combination of weak Fourier sampling and weak Schur sampling fails to identify the hidden subgroup. We also prove tight bounds on how many coset states are required to solve the hidden subgroup problem by weak Schur sampling, and we relate this question to a quantum version of the collision problem.

quant-ph↗