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Language and Automata Theory and Applications

Classical Simulation Complexity of Quantum Branching Programs

Farid Mansurovich Ablayev

Published Jan 1, 2008• 0 citations
Publisher

Abstract

We present classical simulation techniques for measure once quantum branching programs. For bounded error syntactic quantum branching program of width $w$ that computes a function with error $delta$ we present a classical deterministic branching program of the same length and width at most $(1+2/(1-2delta))^{2w}$ that computes the same function. Second technique is a classical stochastic simulation technique for bounded error and unbounded error quantum branching programs. Our result is that it is possible stochastically-classically simulate quantum branching programs with the same length and almost the same width, but we lost bounded error acceptance property.

DOI
10.4230/dagsemproc.07411.3

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We present classical simulation techniques for measure once quantum branching programs. For bounded error syntactic quantum branching program of width www that computes a function with error deltadeltadelta we present a classical deterministic branching program of the same length and width at most (1+2/(1−2delta))2w(1+2/(1-2delta))^{2w}(1+2/(1−2delta))2w that computes the same function. Second technique is a classical stochastic simulation technique for bounded error and unbounded error quantum branching programs. Our result is that it is possible stochastically-classically simulate quantum branching programs with the same length and almost the same width, but we lost bounded error acceptance property.
We present a classical stochastic simulation technique of quantum Branching programs. This technique allows to prove the following relations among complexity classes: PrQP-BP &subuline; PP-BP and BQP-BP &subuline; PP-BP. Here BPP-BP and PP-BP stands for the classes of functions computable with bounded error and unbounded error respectively by stochastic branching program of polynomial size. BQP-BP and PrQP-BP stands the classes of functions computable with bounded error and unbounded error respectively by quantum branching program of polynomial size. Second. We present two different types, of complexity lower bounds for quantum nonuniform automata (OBDDs). We call them "metric" and "entropic" lower bounds in according to proof technique used. We present explicit Boolean functions that show that these lower bounds are tight enough. We show that when considering "almost all Boolean functions" on n variables our entropic lower bounds gives exponential (2c(δ)(n-log n)) lower bound for the width of quantum OBDDs depending on the error δ allowed.
We define a class of stochastic processes based on evolutions and measurements of quantum systems, and consider the complexity of predicting their long term behavior. It is shown that a very general class of decision problems regarding these stochastic processes can be efficiently solved classically in the space-bounded case. The following corollaries are implied by our main result for any space-constructible space bound s satisfying s(n)=/spl Omega/(log n): (i) any space O(s) uniform family of quantum circuit acting on s qubits and consisting of unitary gates and measurement gates defined in a typical way by matrices of algebraic numbers can be simulated by an unbounded error space O(s) ordinary (i.e., fair-coin flipping) probabilistic Turing machine, and hence by space O(s) uniform classical (deterministic) circuits of depth O(s/sup 2/) and size 2/sup 0/(s); (2) any quantum Turing machine running in space s, having arbitrary algebraic transition amplitudes, allowing unrestricted measurements during its computation, and having no restrictions on running time can be simulated by a space O(s) ordinary probabilistic Turing machine in the unbounded error setting. We also obtain the following classical result: any unbounded error probabilistic Turing machine running in space s that allows algebraic probabilities and algebraic cut-point can be simulated by a space O(s) ordinarily probabilistic Turing machine with cut-point 1/2. Our technique for handling algebraic numbers in the above simulations may be of independent interest. It is shown that any real algebraic number can be accurately approximated by a ratio of GapL functions.
In this dissertation, we investigate the computational power of quantum Turing machines operating in bounded space. First, we consider space-bounds that are space-constructible and at least logarithmic in the input size. For such space-bounds, it is shown that quantum Turing machines and probabilistic Turing machines are equivalent in power in the unbounded-error setting, in the sense that each model may simulate the other with at most a constant factor increase in space. From this, it follows that any quantum Turing machine computation can be simulated deterministically with at most a quadratic increase in space, and can be simulated deterministically in time at most exponential in the space-bound. Several other facts regarding quantum complexity classes defined in terms of such space-bounds are also proved. Second, we consider the power of quantum Turing machines restricted to constant space. In this case, we first prove that quantum Turing machines having one-sided error and running in linear time are strictly more powerful than probabilistic Turing machines having either one-sided error or having two-sided bounded error and running in polynomial time. Second, we prove that exact (i.e., accepting with probability 0 or 1) constant-space quantum Turing machines upon which no restrictions on running time are placed can recognize languages that cannot be recognized by any bounded-error constant-space probabilistic Turing machine. ii Acknowledgements I would like to take this opportunity to express my gratitude to a number of people. First, I would like to thank my advisor Eric Bach for all that he has taught me, regarding both theoretical computer science and how to be a researcher in general. His advice and support are greatly appreciated. I would also like to thank Anne Condon for her great advice and for many helpful discussions during my years as a graduate student. This dissertation has benefitted greatly from comments and suggestions by Anne and Eric as well.
This paper investigates the relative power of space-bounded quantum and classical (probabilistic) computational models. The following relationships are proved. 1. Any probabilistic Turing machine (PTM) which runs in space s and which halts absolutely (i.e. halts with certainty after a finite number of steps) can be simulated in space O(s) by a quantum Turing machine (QTM). If the PTM operates with bounded error, then the QTM may be taken to operate with bounded error as well, although the QTM may not halt absolutely in this case. In the unbounded error case, the QTM may be taken to halt absolutely. 2. Any QTM running in space s can be simulated by an unbounded error PTM running in space O(s). No assumptions on the probability of error or Turing time for the QTM are required, but it is assumed that all transition amplitudes of the quantum machine are rational. It follows that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in power. This implies that any space s QTM can be simulated deterministically in space O(s/sup 2/), and further that any (unbounded-error) QTM running in log-space can be simulated in NC/sup 2/ We also consider quantum analogues of nondeterministic and one-sided error probabilistic space-bounded classes, and prove some simple relationships regarding these classes.