Yao's 1995 publication ‘Quantum circuit complexity’ in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science , pp. 352–361, proved that quantum Turing machines and quantum circuits are polynomially equivalent computational models: t ≥ n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quantum circuits with size quadratic in t , and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, which is the more challenging of the two simulation tasks, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t , rather than quadratic depth, and can be extended to variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of method described by Arright, Nesme and Werner in 2011 in Journal of Computer and System Sciences 77 , 372–378. ( doi:10.1016/j.jcss.2010.05.004 ), that allows for the localization of causal unitary evolutions.
Yao's 1995 publication ‘Quantum circuit complexity’ in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science , pp. 352–361, proved that quantum Turing machines and quantum circuits are polynomially equivalent computational models: t ≥ n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quantum circuits with size quadratic in t , and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, which is the more challenging of the two simulation tasks, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t , rather than quadratic depth, and can be extended to variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of method described by Arright, Nesme and Werner in 2011 in Journal of Computer and System Sciences 77 , 372–378. ( doi:10.1016/j.jcss.2010.05.004 ), that allows for the localization of causal unitary evolutions.
We present results on quantum Turing machines and on prover-verifier interactions. \n \n \nIn our work on quantum Turing machines, we continue the line of research opened by Yao (1993), who proved that quantum Turing machines and quantum circuits are polynomially equivalent computational models: t ≥ n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quantum circuits with size quadratic in t, and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We then first revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t, rather than quadratic depth, and can be extended easily to many variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of a method of Arrighi, Nesme, and Werner (2011) that allows for the localization of causal unitary evolutions, involving abstract lemmas that might be of independent interest. \n \n \nWe also consider the more complex extension of our variant to the circuit simulation of multi-tape quantum Turing machines, where our variant provides a circuit with O(t^k) size and O(t^{k-1}) depth for the simulation of t steps of a machine with k tapes. This can be contrasted with the O(t^{k}) depth corresponding to the generalization of Yao's simulation by Nishimura and Ozawa (2002). Our usage of abstract techniques regarding the localization of causal unitary evolutions allows again for a simplification of the algebraic manipulation aspects of the construction. We also discuss the further extension to the case of oracle quantum Turing machines. \n \n \nIn our work on prover-verifier interactions, we first consider a protocol under the name of perfect/conclusive quantum state exclusion. This means to be able to discard with certainty at least one out of n possible quantum state preparations by performing a measurement of the resulting state. When all the preparations correspond to pure states and there are no more of them than their common dimension, it is an open problem whether POVMs give any additional power for this task with respect to projective measurements. This is the case even for the simple case of three states in three dimensions, which is discussed by Caves, Fuchs and Schack (2002) as unsuccessfully tackled. In our work, we give an analytical proof that in this case POVMs do indeed not give any additional power with respect to projective measurements. We also discuss possible generalizations of our work, including an application of Quadratically Constrained Quadratic Programming that might be of special interest. \n \n \nWe additionally consider the problem of quantum hedging, a particular kind of quantum correlation that arises between parallel instances of prover-verifier interactions. M. and Watrous (2012) studied a protocol that exhibited a perfect form of quantum hedging, where the risk for the prover of losing a first game can completely offset the corresponding risk for a second game. We take a step towards a better understanding of this hedging phenomenon by giving a characterization of the prover's optimal behavior for a natural generalization of this protocol. Furthermore, we discuss how the usage of the logarithmic utility principle to analyze prover-verifier interactions could justify further study of quantum hedging.
Understanding which subclasses of quantum circuits are efficiently classically simulable is fundamental to delineating the boundary between classical and quantum computation. In this context, it is well known that certain tasks based on permutation-equivariant unitaries-i.e., n-qubit circuits whose action commutes with the qubit-permuting representation of the symmetric group Sn-can be simulated in polynomial time. However, existing approaches scale as O(n7), and can rapidly become prohibitively expensive. In this work, we introduce a practical algorithm for simulating Sn-equivariant circuits under the assumption that the gate generators are at most k-local, with k∈O(1). The resulting method runs in O(nω+1) time for constant depth, where ω is the matrix multiplication exponent, significantly lowering the polynomial degree compared to existing techniques. Finally, we numerically validate this scaling by simulating the dynamical evolution of the Lipkin-Meshkov-Glick model, and show that for n=512 spins, a standard laptop can compute the concurrence of the evolved state in under two minutes.
Abstract In the past four decades, the notion of quantum polynomial-time computability has been mathematically modeled by quantum Turing machines as well as quantum circuits. This paper seeks the third model, which is a quantum analogue of the schematic (inductive or constructive) definition of (primitive) recursive functions. For quantum functions mapping finite-dimensional Hilbert spaces to themselves, we present such a schematic definition, composed of a small set of initial quantum functions and a few construction rules that dictate how to build a new quantum function from the existing ones. We prove that our schematic definition precisely characterizes all functions that can be computable with high success probabilities on well-formed quantum Turing machines in polynomial time, or equivalently uniform families of polynomial-size quantum circuits. Our new, schematic definition is quite simple and intuitive and, more importantly, it avoids the cumbersome introduction of the well-formedness condition imposed on a quantum Turing machine model as well as of the uniformity condition necessary for a quantum circuit model. Our new approach can further open a door to the descriptional complexity of quantum functions, to the theory of higher-type quantum functionals, to the development of new first-order theories for quantum computing, and to the designing of programming languages for real-life quantum computer
The model of local Turing machines is introduced, including classical and quantum ones, in the framework of matrix-product states. The locality refers to the fact that at any instance of the computation the heads of a Turing machine have definite locations. The local Turing machines are shown to be equivalent to the corresponding circuit models and standard models of Turing machines by simulation methods. This work reveals the fundamental connection between tensor-network states and information processing.
Quantum computation was proposed as an alternate computing process in the late-20th century, when classical computing models faced limitation in solving quantum mechanical problems. Major breakthroughs came in the 1990s, when quantum algorithms for integer factoring and black-box searching were invented that are faster than the best classical algorithms known even today. In this thesis, we look at quantum circuits with limited depth and size and analyse their power and limitations.
Computer Science is about the study of computational problems, many of which are motivated by real-world applications. We are interested as much in an actual method to solve a problem as in its unsolvability. In computational complexity theory we separate out the actual problem from the computing apparatus (e.g. a Turing machine, an AND-OR circuit) and analyse the hardness of the computation, due both to the intrinsic complexity of the problem and to the underlying model.
In this thesis, we consider quantum circuits built with a universal set of gates and show that non-constant depth is required to compute a particular function (parity) and other similar functions. The parity function, extensively studied previously, was shown to take non-constant depth for classical circuits too, thus providing an evidence of an inherent hardness in the function itself. On the other hand we show that the quantum circuit model is rich enough to build a universal circuit (one circuit that can simulate all circuits) that has roughly the same depth or size as the simulated circuit. We also consider a fault model in which some of the inputs to a gate might not be connected to the gate's output. We ask if we can detect the faulty inputs by testing the gate on different inputs. Our goal is to minimise the number of tests. We show that this is very hard for general gates but for certain gates it can be provably easier to detect faulty inputs. We even show that quantum algorithms can provably outperform classical algorithms for these gates. These results give us a sense of wherein lies the power of quantum computation and quantum circuits in particular.
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.