Complexity of decoupling and time-reversal for n spins with pair-interactions: Arrow of time in quantum control

Well-known Nuclear Magnetic Resonance experiments show that the time evolution according to (truncated) dipole-dipole interactions between n spins can be inverted by simple pulse sequences. Independent of n, the reversed evolution is only two times slower than the original one. Here we consider more general spin-spin couplings with long range. We prove that some are considerably more complex to invert since the number of required time steps and the slow-down of the reversed evolutions are necessarily of the order n. Furthermore, the spins have to be addressed separately. We show for which values of the coupling parameters the phase transition between simple and complex time-reversal schemes occurs.Comment: Completely rewritten, new lower bounds on the number of time steps, applications and references adde

Estimating Jones and HOMFLY polynomials with One Clean Qubit

The Jones and HOMFLY polynomials are link invariants with close connections to quantum computing. It was recently shown that finding a certain approximation to the Jones polynomial of the trace closure of a braid at the fifth root of unity is a complete problem for the one clean qubit complexity class. This is the class of problems solvable in polynomial time on a quantum computer acting on an initial state in which one qubit is pure and the rest are maximally mixed. Here we generalize this result by showing that one clean qubit computers can efficiently approximate the Jones and single-variable HOMFLY polynomials of the trace closure of a braid at any root of unity.Comment: 22 pages, 11 figures, revised in response to referee comment

Efficient decoupling schemes with bounded controls based on Eulerian orthogonal arrays

The task of decoupling, i.e., removing unwanted interactions in a system Hamiltonian and/or couplings with an environment (decoherence), plays an important role in controlling quantum systems. There are many efficient decoupling schemes based on combinatorial concepts like orthogonal arrays, difference schemes and Hadamard matrices. So far these (combinatorial) decoupling schemes have relied on the ability to effect sequences of instantaneous, arbitrarily strong control Hamiltonians (bang-bang controls). To overcome the shortcomings of bang-bang control Viola and Knill proposed a method called Eulerian decoupling that allows the use of bounded-strength controls for decoupling. However, their method was not directly designed to take advantage of the composite structure of multipartite quantum systems. In this paper we define a combinatorial structure called an Eulerian orthogonal array. It merges the desirable properties of orthogonal arrays and Eulerian cycles in Cayley graphs (that are the basis of Eulerian decoupling). We show that this structure gives rise to decoupling schemes with bounded-strength control Hamiltonians that can be applied to composite quantum systems with few body Hamiltonians and special couplings with the environment. Furthermore, we show how to construct Eulerian orthogonal arrays having good parameters in order to obtain efficient decoupling schemes.Comment: 8 pages, revte

Simulating Hamiltonians in Quantum Networks: Efficient Schemes and Complexity Bounds

We address the problem of simulating pair-interaction Hamiltonians in n node quantum networks where the subsystems have arbitrary, possibly different, dimensions. We show that any pair-interaction can be used to simulate any other by applying sequences of appropriate local control sequences. Efficient schemes for decoupling and time reversal can be constructed from orthogonal arrays. Conditions on time optimal simulation are formulated in terms of spectral majorization of matrices characterizing the coupling parameters. Moreover, we consider a specific system of n harmonic oscillators with bilinear interaction. In this case, decoupling can efficiently be achieved using the combinatorial concept of difference schemes. For this type of interactions we present optimal schemes for inversion.Comment: 19 pages, LaTeX2

Efficient quantum processing of ideals in finite rings

Suppose we are given black-box access to a finite ring R, and a list of generators for an ideal I in R. We show how to find an additive basis representation for I in poly(log |R|) time. This generalizes a recent quantum algorithm of Arvind et al. which finds a basis representation for R itself. We then show that our algorithm is a useful primitive allowing quantum computers to rapidly solve a wide variety of problems regarding finite rings. In particular we show how to test whether two ideals are identical, find their intersection, find their quotient, prove whether a given ring element belongs to a given ideal, prove whether a given element is a unit, and if so find its inverse, find the additive and multiplicative identities, compute the order of an ideal, solve linear equations over rings, decide whether an ideal is maximal, find annihilators, and test the injectivity and surjectivity of ring homomorphisms. These problems appear to be hard classically.Comment: 5 page

Entanglement Capacity of Nonlocal Hamiltonians : A Geometric Approach

We develop a geometric approach to quantify the capability of creating entanglement for a general physical interaction acting on two qubits. We use the entanglement measure proposed by us for $N$-qubit pure states (PRA \textbf{77}, 062334 (2008)). Our procedure reproduces the earlier results (PRL \textbf{87}, 137901 (2001)). The geometric method has the distinct advantage that it gives an experimental way to monitor the process of optimizing entanglement production.Comment: 8 pages, 1 figure

Testing quantum expanders is co-QMA-complete

A quantum expander is a unital quantum channel that is rapidly mixing, has only a few Kraus operators, and can be implemented efficiently on a quantum computer. We consider the problem of estimating the mixing time (i.e., the spectral gap) of a quantum expander. We show that this problem is co-QMA-complete. This has applications to testing randomized constructions of quantum expanders, and studying thermalization of open quantum systems