On the complexity of simulating space-bounded quantum computations

Tomoyuki Yamakami

John Watrous

There are a number of questions in quantum complexity that have been resolved in the time-bounded setting, but remain open in the space-bounded setting. For example, it is not currently known if space-bounded probabilistic computations can be simulated by space-bounded quantum machines without allowing measurements during the computation, while it is known that an analogous statement holds in the time-bounded case. A more general question asks if measurements during a quantum computation can allow for more space-efficient solutions to certain problems. In this paper we show that space-bounded quantum Turing machines can efficiently simulate a limited class of random processes-random walks on undirected graphs-without relying on measurements during the computation. By means of such simulations, it is demonstrated that the undirected graph connectivity problem for regular graphs can be solved by one-sided error quantum Turing machines that run in logspace and require a single measurement at the end of their computations. It follows that symmetric logspace is contained in the quantum analogue of randomized logspace, i.e., SL/spl sube/QR/sub H/L.

Richard Jozsa

Harry Buhrman, Folkertsma, Marten, Ian Mertz +4 more

Space complexity is a key field of study in theoretical computer science. In the quantum setting there are clear motivations to understand the power of space-restricted computation, as qubits are an especially precious and limited resource. Recently, a new branch of space-bounded complexity called catalytic computing has shown that reusing space is a very powerful computational resource, especially for subroutines that incur little to no space overhead. While quantum catalysis in an information theoretic context, and the power of "dirty" qubits for quantum computation, has been studied over the years, these models are generally not suitable for use in quantum space-bounded algorithms, as they either rely on specific catalytic states or destroy the memory being borrowed. We define the notion of catalytic computing in the quantum setting and show a number of initial results about the model. First, we show that quantum catalytic logspace can always be computed quantumly in polynomial time; the classical analogue of this is the largest open question in catalytic computing. This also allows quantum catalytic space to be defined in an equivalent way with respect to circuits instead of Turing machines. We also prove that quantum catalytic logspace can simulate log-depth threshold circuits, a class which is known to contain (and believed to strictly contain) quantum logspace, thus showcasing the power of quantum catalytic space. Finally we show that both unitary quantum catalytic logspace and classical catalytic logspace can be simulated in the one-clean qubit model.

孝志 三原

John Watrous, Eric Bach

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.