QIC Abstracts

 Vol.15 No.1&12, January 1, 2015

Research Articles:

Improving quantum algorithms for quantum chemistry (pp0001-0021)
Matthew B. Hastings, Dave Wecker, Bela Bauer, and Matthias Troyer
We present several improvements to the standard Trotter-Suzuki based algorithms used in the simulation of quantum chemistry on a quantum computer. First, we modify how Jordan-Wigner transformations are implemented to reduce their cost from linear or logarithmic in the number of orbitals to a constant. Our modification does not require additional ancilla qubits. Then, we demonstrate how many operations can be parallelized, leading to a further linear decrease in the parallel depth of the circuit, at the cost of a small constant factor increase in number of qubits required. Thirdly, we modify the term order in the Trotter-Suzuki decomposition, significantly reducing the error at given Trotter-Suzuki timestep. A final improvement modifies the Hamiltonian to reduce errors introduced by the non-zero Trotter-Suzuki timestep. All of these techniques are validated using numerical simulation and detailed gate counts are given for realistic molecules.

A model of quantum-von Neumann hybrid cellular automata: principles and simulation of quantum coherent superposition and dehoerence in cytoskeletal microtubules (pp0022-0036)
Manuel Alfonseca, Alfonso Ortega, Marina de la Cruz, Stuart R. Hameroff, and Rafael Lahoz-Beltra
Although experimental evidence suggests the influence of quantum effects in living organisms, one of the most critical problems in quantum biology is the explanation of how those effects that take place in a microscopic level can manifest in the macroscopic world of living beings. At present, quantum decoherence associated with the wave function collapse is one of the most accepted mechanisms explaining how the classical world of living beings emerges from the quantum world. Whatever the cause of wave function collapse, there exist biological systems where a biological function arises as a result of this collapse (e.g. birds navigation, plants photosynthesis, sense of smell, etc.), as well as the opposite examples (e.g. release of energy from ATP molecules at actomyosin muscle) where a biological function takes place in a quantum coherent environment. In this paper we report the modelling and simulation of quantum coherent superposition in cytoskeletal microtubules including decoherence, thus the effect of the collapse of the microtubule coherent state wave function. Our model is based on a new class of hybrid cellular automata (QvN), capable of performing as either a quantum cellular automata (QCA) or as a classical von Neumann automata (CA). These automata are able to simulate the transition or reduction from a quantum microscopic level with superposition of several quantum states, to a macroscopic level with a single stable state. Our results illustrate the significance of quantum biology explaining the emergence of some biological functions. We believe that in the future quantum biology will have a deep effect on the design of new devices, e.g. quantum hardware, in electrical engineering.

Detection loophole attacks on semi-device-independent quantum and classical protocols (pp0037-0049)
Michele Dall'Arno, Elsa Passaro, Rodrigo Gallego, Marcin Pawlowski, and Antonio Acin
Semi-device-independent quantum protocols realize information tasks -- e.g. secure key distribution, random access coding, and randomness generation -- in a scenario where no assumption on the internal working of the devices used in the protocol is made, except their dimension. These protocols offer two main advantages: first, their implementation is often less demanding than fully-device-independent protocols. Second, they are more secure than their device-dependent counterparts. Their classical analogous is represented by random access codes, which provide a general framework for describing one-sided classical communication tasks. We discuss conditions under which detection inefficiencies can be exploited by a malicious provider to fake the performance of semi-device-independent quantum and classical protocols -- and how to prevent it.

A limit theorem for a 3-period time-dependent quantum walk (pp0050-0060)
F. Alberto Grunbaum and Takuya Machida
We consider a discrete-time 2-state quantum walk on the line. The state of the quantum walker evolves according to a rule which is determined by a coin-flip operator and a position-shift operator.
In this paper we take a 3-periodic time evolution as the rule. For such a quantum walk, we get a limit distribution which expresses the asymptotic behavior of the walker after a long time. The limit distribution is different from that of a time-independent quantum walk or a 2-period time-dependent quantum walk. We give some analytical results and then consider a number of variants of our model and indicate the result of simulations for these ones.

Group theoretic, Lie algebraic and Jordan algebraic formulations of the SIC existence problem (pp0061-0094)
D. M. Appleby, Christopher A. Fuchs, and Huangjun Zhu
Although symmetric informationally complete positive operator valued measures (SIC POVMs, or SICs for short) have been constructed in every dimension up to 67, a general existence proof remains elusive. The purpose of this paper is to show that the SIC existence problem is equivalent to three other, on the face of it quite different problems. Although it is still not clear whether these reformulations of the problem will make it more tractable, we believe that the fact that SICs have these connections to other areas of mathematics is of some intrinsic interest. Specifically, we reformulate the SIC problem in terms of (1) Lie groups, (2) Lie algebras and (3) Jordan algebras (the second result being a greatly strengthened version of one previously obtained by Appleby, Flammia and Fuchs). The connection between these three reformulations is non-trivial: It is not easy to demonstrate their equivalence directly, without appealing to their common equivalence to SIC existence. In the course of our analysis we obtain a number of other results which may be of some independent interest.

Controlling the coherence in a pure dephasing model for arbitrary prescibed time span (pp0095-0104)
Lucio Fassarella
We present an open-loop unitary strategy to control the coherence in a pure dephasing model (related to the phase-flip channel) that is able to recover, for whatever prescribed time span, the initial coherence at the end of the control process. The strategy's key idea is to steer the quantum state to the subset of invariant states and keep it there the necessary time, using a fine tuned control Hamiltonian.

Time-averaged limit measure of the Wojcik model (pp0105-0133)
Takako Endo and Norio Konno
We investigate ``the Wojcik model" introduced and studied by Wojcik et al. \cite{wojcik}, which is a one-defect quantum walk (QW) having a single phase at the origin. They reported that giving a phase at one point causes an astonishing effect for localization. Three types of measures have important roles in the study of QWs: time-averaged limit measure, weak limit measure, and stationary measure. The first two measures imply a coexistence of localized behavior and the ballistic spreading in the QW. As Konno et al. \cite{segawa} suggested, the time-averaged limit and stationary measures are closely related to each other for some QWs with one defect. In this paper, we focus on a relation between the two measures for the Wojcik model. The stationary measure was already obtained by our previous work \cite{watanabe}. Here, we get the time-averaged limit measure by several methods. Our results show that the stationary measure is a special case of the time-averaged limit measure.

Polyestimate: a library for near-instantaneous surface code analysis (pp0134-0144)
Austin G. Fowler
The surface code is highly practical, enabling arbitrarily reliable quantum computation given a 2-D nearest-neighbor coupled array of qubits with gate error rates below approximately 1\%. We describe an open source library, Polyestimate, enabling a user with no knowledge of the surface code to specify realistic physical quantum gate error models and obtain logical error rate estimates. Functions allowing the user to specify simple depolarizing error rates for each gate have also been included. Every effort has been made to make this library user-friendly. Polyestimate provides data essentially instantaneously that previously required hundreds to thousands of hours of simulation, statements which we discuss and make precise. This advance has been made possible through careful analysis of the error structure of the surface code and extensive pre-simulation.

Minimum weight perfect matching of fault-tolerant topological quantum error correction in average O(1) parallel time (pp0145-0158)
Austin G. Fowler
Consider a 2-D square array of qubits of extent $L\times L$. We provide a proof that the minimum weight perfect matching problem associated with running a particular class of topological quantum error correction codes on this array can be exactly solved with a 2-D square array of classical computing devices, each of which is nominally associated with a fixed number $N$ of qubits, in constant average time per round of error detection independent of $L$ provided physical error rates are below fixed nonzero values, and other physically reasonable assumptions. This proof is applicable to the fully fault-tolerant case only, not the case of perfect stabilizer measurements.

Efficient Clifford+T approximation of single-qubit operators (pp0159-0180)
Peter Selinger
We give an efficient randomized algorithm for approximating an arbitrary element of $\SU(2)$ by a product of Clifford+$T$ operators, up to any given error threshold $\epsilon>0$. Under a mild hypothesis on the distribution of primes, the algorithm's expected runtime is polynomial in $\log(1/\epsilon)$. If the operator to be approximated is a $z$-rotation, the resulting gate sequence has $T$-count $K+4\log_2(1/\epsilon)$, where $K$ is approximately equal to $10$. We also prove a worst-case lower bound of $K+4\log_2(1/\epsilon)$, where $K=-9$, so that our algorithm is within an additive constant of optimal for certain $z$-rotations. For an arbitrary member of $\SU(2)$, we achieve approximations with $T$-count $K+12\log_2(1/\epsilon)$. By contrast, the Solovay-Kitaev algorithm achieves $T$-count $O(\log^c(1/\epsilon))$, where $c$ is approximately $3.97$.

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