Quantum Computers and Decoherence: Exorcising the Demon from the Machine

Decoherence is the main obstacle to the realization of quantum computers. Until recently it was thought that quantum error correcting codes are the only complete solution to the decoherence problem. Here we present an alternative that is based on a combination of a decoherence-free subspace encoding and the application of strong and fast pulses: ``encoded recoupling and decoupling'' (ERD). This alternative has the advantage of lower encoding overhead (as few as two physical qubits per logical qubit suffice), and direct application to a number of promising proposals for the experimental realization of quantum computers.Comment: 15 pages, no figures. Invited contribution to the proceedings of the SPIE Conference on Fluctuations and Noise. Section 8 contains a new result: how to eliminate off-resonant transitions induced by generic "bang-bang" pulses, by using a special type of "bang-bang" pulse

Universal computation by multi-particle quantum walk

A quantum walk is a time-homogeneous quantum-mechanical process on a graph defined by analogy to classical random walk. The quantum walker is a particle that moves from a given vertex to adjacent vertices in quantum superposition. Here we consider a generalization of quantum walk to systems with more than one walker. A continuous-time multi-particle quantum walk is generated by a time-independent Hamiltonian with a term corresponding to a single-particle quantum walk for each particle, along with an interaction term. Multi-particle quantum walk includes a broad class of interacting many-body systems such as the Bose-Hubbard model and systems of fermions or distinguishable particles with nearest-neighbor interactions. We show that multi-particle quantum walk is capable of universal quantum computation. Since it is also possible to efficiently simulate a multi-particle quantum walk of the type we consider using a universal quantum computer, this model exactly captures the power of quantum computation. In principle our construction could be used as an architecture for building a scalable quantum computer with no need for time-dependent control

Fault-tolerant gates via homological product codes

A method for the implementation of a universal set of fault-tolerant logical gates is presented using homological product codes. In particular, it is shown that one can fault-tolerantly map between different encoded representations of a given logical state, enabling the application of different classes of transversal gates belonging to the underlying quantum codes. This allows for the circumvention of no-go results pertaining to universal sets of transversal gates and provides a general scheme for fault-tolerant computation while keeping the stabilizer generators of the code sparse.Comment: 11 pages, 3 figures. v2 (published version): quantumarticle documentclass, expanded discussion on the conditions for a fault tolerance threshol

Selective and Efficient Quantum Process Tomography

In this paper we describe in detail and generalize a method for quantum process tomography that was presented in [A. Bendersky, F. Pastawski, J. P. Paz, Physical Review Letters 100, 190403 (2008)]. The method enables the efficient estimation of any element of the $\chi$--matrix of a quantum process. Such elements are estimated as averages over experimental outcomes with a precision that is fixed by the number of repetitions of the experiment. Resources required to implement it scale polynomically with the number of qubits of the system. The estimation of all diagonal elements of the $\chi$--matrix can be efficiently done without any ancillary qubits. In turn, the estimation of all the off-diagonal elements requires an extra clean qubit. The key ideas of the method, that is based on efficient estimation by random sampling over a set of states forming a 2--design, are described in detail. Efficient methods for preparing and detecting such states are explicitly shown.Comment: 9 pages, 5 figure

Entropy Cones and Entanglement Evolution for Dicke States

The $N$-qubit Dicke states $|D^N_k\rangle$, of Hamming-weight $k$, are a class of entangled states which play an important role in quantum algorithm optimization. We present a general calculation of entanglement entropy in Dicke states, which we use to describe the $|D^N_k\rangle$ entropy cone. We demonstrate that all $|D^N_k\rangle$ entropy vectors emerge symmetrized, and use this to define a min-cut protocol on star graphs which realizes $|D^N_k\rangle$ entropy vectors. We identify the stabilizer group for all $|D^N_k\rangle$, under the action of the $N$-qubit Pauli group and two-qubit Clifford group, which we use to construct $|D^N_k\rangle$ reachability graphs. We use these reachability graphs to analyze and bound the evolution of $|D^N_k\rangle$ entropy vectors in Clifford circuits.Comment: 31 pages, 13 figures, 1 Mathematica packag

Hamiltonian quantum simulation with bounded-strength controls

We propose dynamical control schemes for Hamiltonian simulation in many-body quantum systems that avoid instantaneous control operations and rely solely on realistic bounded-strength control Hamiltonians. Each simulation protocol consists of periodic repetitions of a basic control block, constructed as a suitable modification of an "Eulerian decoupling cycle," that would otherwise implement a trivial (zero) target Hamiltonian. For an open quantum system coupled to an uncontrollable environment, our approach may be employed to engineer an effective evolution that simulates a target Hamiltonian on the system, while suppressing unwanted decoherence to the leading order. We present illustrative applications to both closed- and open-system simulation settings, with emphasis on simulation of non-local (two-body) Hamiltonians using only local (one-body) controls. In particular, we provide simulation schemes applicable to Heisenberg-coupled spin chains exposed to general linear decoherence, and show how to simulate Kitaev's honeycomb lattice Hamiltonian starting from Ising-coupled qubits, as potentially relevant to the dynamical generation of a topologically protected quantum memory. Additional implications for quantum information processing are discussed.Comment: 24 pages, 5 color figure