Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

Abstract

We consider quantum computational models defined via a Lie-algebraic theory. In these models, specified initial states are acted on by Lie-algebraic quantum gates and the expectation values of Lie algebra elements are measured at the end. We show that these models can be efficiently simulated on a classical computer in time polynomial in the dimension of the algebra, regardless of the dimension of the Hilbert space where the algebra acts. Similar results hold for the computation of the expectation value of operators implemented by a gatesequence. We introduce a Lie-algebraic notion of generalized mean-field Hamiltonians and show that they are efficiently (exactly) solvable by means of a Jacobi-like diagonalization method. Our results generalize earlier ones on fermionic linear optics computation and provide insight into the source of the power of the conventional model of quantum computation. Quantum models of computation are widely believed to be more powerful than classical ones. Although this has been shown to be true in a few cases, it is still important to determine when a quantum algorithm for a given problem is more resource efficient than any classical one, or, conversely, when a classical algorithm is just as efficient as any quantum counterpart. In general, one needs to know whether it is worth investing in building a quantum computer (QC) and what is required for success. In this paper, we show close connections between these issues and the efficient (or exact) solvability of Hamiltonians. In particular, we show that a class of quantum models we call generalized mean-field Hamiltonians (GMFHs) [1] is efficiently solvable and furthermore does not provide a stronger-than-classical model of computation: A quantum device engineered to have dynamical gates generated by Hamiltonians from such a set cannot directly simulate universal efficient quantum computation and can be efficiently simulated by a classical computer (CC).An algorithm is a sequence of elementary instructions that solves instances of a problem. It is said to be efficient if the resources required to solve problem instances of size N are polynomial in N (poly(N )) resources. Typically, the size of a problem instance is the number of bits required to represent it, and the relevant resources are time and space. In the last few years it has been shown that many pure-state quantum algorithms can be efficiently simulated on a CC when the extent of entanglement is limited (e.g., [2,3]) or when the quantum gates available are far from allowing us to build a set of universal gates [4,5,6]. Here, we focus on a Lie algebraic analysis to obtain other situations where quantum algorithms can be efficiently simulated by CCs. The so-called generalized coherent states (GCSs) [7] play a decisive role in our analysis.The algorithms considered here make use of the Liealgebraic model of quantum computing (LQC). An LQC algorithm begins with the specification of a semisimple, compact M -dimensional real Lie algebraĥ of skew-Hermitian operators acting on a finite-...

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