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Journal of Applied and Industrial Mathematics

On simulating the quantum and classical branching programs

Aida Gainutdinova

Published Mar 1, 2007• 1 citation
Publisher

Abstract

The complexity classes defined on the basis of branching programs are considered. Some basic relations are established between the complexity classes defined by the probabilistic and quantum branching programs (measure-once, as well as measure-many), computing with bounded or unbounded error. To prove these relations, we developed a method of “linear simulation” of a quantum branching program and a method of “quantum simulation” of a probabilistic branching program.

DOI
10.1134/s1990478907010048

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The complexity classes defined on the basis of branching programs are considered. Some basic relations are established between the complexity classes defined by the probabilistic and quantum branching programs (measure-once, as well as measure-many), computing with bounded or unbounded error. To prove these relations, we developed a method of “linear simulation” of a quantum branching program and a method of “quantum simulation” of a probabilistic branching program.
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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.
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.
We investigate the branching program complexity of quantum hashing. We consider a quantum hash function that maps elements of a finite field into quantum states. We require that this function is preimage-resistant and collision-resistant.We consider two complexity measures for Quantum Branching Programs (QBP): a number of qubits and a number of compu-tational steps. We show that the quantum hash function can be computed efficiently. Moreover, we prove that such QBP construction is optimal. That is, we prove lower bounds that match the constructed quantum hash function computation.
We show that classical and quantum Kolmogorov complexity of binary words agree up to an additive constant. Both complexities are defined as the minimal length of any (classical resp. quantum) computer program that outputs the corresponding word. It follows that quantum complexity is an extension of classical complexity to the domain of quantum states. This is true even if we allow a small probabilistic error in the quantum computer’s output. We outline a mathematical proof of this statement, based on some analytical estimates and a classical program for the simulation of a universal quantum computer.