Eigenstates of an operating quantum computer: hypersensitivity to static imperfections

We study the properties of eigenstates of an operating quantum computer which simulates the dynamical evolution in the regime of quantum chaos. Even if the quantum algorithm is polynomial in number of qubits nq, it is shown that the ideal eigenstates become mixed and strongly modified by static imperfections above a certain threshold which drops exponentially with nq . Above this threshold the quantum eigenstate entropy grows linearly with nq but the computation remains reliable during a time scale which is polynomial in the imperfection strength and in nq . PACS. 03.67.Lx Quantum computation – 05.45.Mt Semiclassical chaos (“quantum chaos”) – 24.10.Cn Many-body theory Feynman suggested that a quantum computer could simulate quantum mechanical systems exponentially faster than a classical computer [1] while Shor significantly extended this class by his ground-breaking algorithm for integer factorization [2]. More recently, a few quantum algorithms which achieve the exponential speedup have been developed for various quantum and classical physical systems, ranging from some many-body problems [3] to spin lattices [4], and models of quantum chaos [5–7]. It is important to study the stability of these algorithms in the presence of concrete models of decoherence and quantum computer imperfections [8,9]. The first investigations have shown a certain stability of quantum evolution and algorithms with respect to decoherence effects [10], noisy gates [11–13], and static imperfections [7,9]. These studies have focused on the fidelity of quantum computation as a function of time during the realization of a given quantum algorithm. In this paper, we study the properties of the eigenstates of an operating quantum computer in the presence of static imperfections. The computer is simulating efficiently the time evolution of a dynamical quantum system described by the sawtooth map [7]. We focus on the regime of quantum ergodicity, in which eigenfunctions are given by a complex superposition of a large number of quantum register states. In this regime, the effect of a perturbation is enhanced by a factor which is exponential in the numa e-mail: [email protected] b UMR 5626 du CNRS ber of qubits. This phenomenon has close links with the enormous enhancement of weak interactions in heavy nuclei [14]. In the following we illustrate this general effect for the case of static imperfections in a realistic model of quantum computer hardware. The classical sawtooth map is given by n = n+ k(θ − π), θ = θ + Tn, (1) where (n, θ) are conjugated action-angle variables (0 ≤ θ < 2π), and the bars denote the variables after one map step. Introducing the rescaled momentum variable p = Tn, one can see that the classical dynamics depends only on the single parameter K = kT , so that the motion is stable for −4 < K < 0 and completely chaotic for K < −4 and K > 0. The quantum evolution for one map iteration is described by a unitary operator Û acting on the wave function ψ: ψ = Ûψ = e−iTn̂ 2/2eik(θ̂−π) 2/2ψ, (2) where n̂ = −i∂/∂θ (we set ~ = 1). The classical limit corresponds to k →∞, T → 0, and K = kT = const . In this paper, we study the quantum sawtooth map (2) in the regime of quantum ergodicity, with K = √ 2, −π ≤ p < π (torus geometry). The classical limit is obtained by increasing the number of qubits nq = log2N (N number of levels), with T = 2π/N (k = K/T , −N/2 ≤ n < N/2). The quantum algorithm [7] simulates with exponential efficiency the quantum dynamics (2) using a register of nq 294 The European Physical Journal D qubits. It is based on the forward/backward quantum Fourier transform [15] between the θ and n representations and requires 2nq Hadamard gates and 3nq−nq controlledphase-shift gates per map iteration [7]. Following [9], we model the quantum computer hardware as an one-dimensional array of qubits (spin halves) with static imperfections, i.e. fluctuations in the individual qubit energies and residual short-range inter-qubit couplings. The model is described by the many-body Hamiltonian