We define a complexity class for randomized algorithms with one-sided error that is exactly equal to a constant (unlike the usual definitions, in which the error is only bounded above or below by a constant). We show that the corresponding quantum classes (one each for a different error probability) are in fact all equivalent to each other and to EQP, the quantum analogue of P. The technique used is a form of quantum amplitude amplification.
We define a complexity class for randomized algorithms with one-sided error that is exactly equal to a constant (unlike the usual definitions, in which the error is only bounded above or below by a constant). We show that the corresponding quantum classes (one each for a different error probability) are in fact all equivalent to each other and to EQP, the quantum analogue of P. The technique used is a form of quantum amplitude amplification.
Amplitude amplification is a central tool used in Grover's quantum search algorithm and has been used in various forms in numerous quantum algorithms since then. It has been shown to completely eliminate one-sided error of quantum search algorithms where input is accessed in the form of black-box queries. We generalize amplitude amplification for two-sided error quantum algorithm for decision problems in the familiar form where input is accessed in the form of initial states of quantum circuits and where arbitrary projective measurements may be used to ascertain success or failure. This generalization allows us to derive interesting applications of amplitude amplification for distinguishing between two given distributions based on their samples, detection of faults in quantum circuits and eliminating error of one and two-sided quantum algorithms with exact errors.
We present characterisations of "exact" gap-definable classes, in terms of indeterministic models of computation which slightly modify the standard model of quantum computation. This follows on work of Aaronson [arXiv:quant-ph/0412187], who shows that the counting class PP can be characterised in terms of bounded-error "quantum" algorithms which use invertible (and possibly non-unitary) transformations, or postselections on events of non-zero probability. Our work considers similar modifications of the quantum computational model, but in the setting of exact algorithms, and algorithms with zero error and constant success probability. We show that the gap-definable counting classes [J. Comput. Syst. Sci. 48 (1994), p.116] which bound exact and zero-error quantum algorithms can be characterised in terms of "quantum-like" algorithms involving nonunitary gates, and that postselection and nonunitarity have equivalent power for exact quantum computation only if these classes collapse.
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.
We present a technique to reduce the error probability of quantum algorithms that determine whether its input has a specified property of interest. The standard process of reducing this error is statistical processing of the results of multiple independent executions of an algorithm. Denoting by $\ensuremath{\rho}$ an upper bound of this probability (wlog; assume $\ensuremath{\rho}\ensuremath{\le}\frac{1}{2}$), classical techniques require $O(\frac{\ensuremath{\rho}}{{[(1\ensuremath{-}\ensuremath{\rho})\ensuremath{-}\ensuremath{\rho}]}^{2}})$ executions to reduce the error to a negligible constant. We investigate when and how quantum algorithmic techniques like amplitude amplification and estimation may reduce the number of executions. On one hand, the former idea does not directly benefit algorithms that can err on both yes and no answers and the number of executions in the latter approach is $O(\frac{1}{(1\ensuremath{-}\ensuremath{\rho})\ensuremath{-}\ensuremath{\rho}})$. We propose a quantum approach called amplitude separation that combines both these approaches and achieves $O(\frac{1}{\sqrt{1\ensuremath{-}\ensuremath{\rho}}\ensuremath{-}\sqrt{\ensuremath{\rho}}})$ executions, which betters existing approaches when the errors are high. In the multiple-weight decision problem, the input is an n-bit Boolean function f() given as a black box and the objective is to determine the number of x for which f(x)=1, denoted wt(f), given some possible values w1,...,wk for wt(f). When our technique is applied to this problem, we obtain the correct answer, maybe with a negligible error, using O(log2k2n) calls to f(), which shows a quadratic speedup over classical approaches and currently known quantum algorithms.
Classical approximation complexity asks what solution quality can be guaranteed with polynomial-time computation. The classes APX, PTAS, and FPTAS distinguish a fixed approximation ratio, approximation to any fixed accuracy, and approximation schemes whose running time is also polynomial in inverse accuracy. Their randomized counterparts are R-APX, R-PTAS, and R-FPTAS. We define bounded-error quantum counterparts BQ-APX, BQ-PTAS, and BQ-FPTAS. Membership requires a uniform quantum algorithm that, on every input, returns a feasible classical solution achieving at least the claimed approximation ratio with probability at least 2/3. Scores (objective values) must be efficiently classically computable. Running time includes parameter selection, preparation, measurement, decoding, and repetition. Many quantum optimization methods are used heuristically, and high benchmark scores alone do not establish these guarantees. We further establish a conditional hierarchy for logarithmic, polynomial, and exponential approximation factors.
Assuming NP ⊈ BQP, the quantum classes form a strict hierarchy. Problems based on prime factorization and discrete logarithms give conditional quantum-classical separations. Certified Maximum Order has an exact quantum algorithm, while any randomized polynomial-time algorithm guaranteeing at least an inverse-polynomial approximation ratio on every input would yield efficient factoring. Discrete-Logarithm Fitting has an exact quantum algorithm and a deterministic one-half approximation, but any fixed improvement over one half would give a randomized polynomial-time algorithm for the safe-prime discrete logarithm problem. Our results show, under explicit complexity assumptions, that quantum computation can improve worst-case approximation guarantees. A quantum-classical gap for common problems such as MaxCut or MaxSAT remains open.
We use the venerable "fooling set" method to prove new lower bounds on the quantum communication complexity of various functions. Let f:X x Y-->{0,1} be a Boolean function, fool^1(f) its maximal fooling set size among 1-inputs, Q_1^*(f) its one-sided error quantum communication complexity with prior entanglement, and NQ(f) its nondeterministic quantum communication complexity (without prior entanglement; this model is trivial with shared randomness or entanglement). Our main results are the following, where logs are to base 2: * If the maximal fooling set is "upper triangular" (which is for instance the case for the equality, disjointness, and greater-than functions), then we have Q_1^*(f)>=(1/2)log fool^1(f) - 1/2, which is essentially optimal by superdense coding. No super-constant lower bound for equality seems to follow from earlier techniques. * For all f we have Q_1^*(f)>=(1/4)log fool^1(f) - 1/2, which is optimal up to a factor of 2. * NQ(f)>=log \fool^1(f)/2 + 1. We do not know if the factor 1/2 is needed in this result, but it cannot be replaced by 1: we give an example where NQ(f)~0.613 log fool^1(f).