QIC Abstracts

 Vol.16 No.11&12, September 1, 2016

Research Articles:

Optimal ancilla-free Clifford+$T$ approximation of z-rotations (pp0901-0953)
          
Neil J. Ross and Peter Selinger
We consider the problem of approximating arbitrarysingle-qubit $z$-rotations by ancilla-free Clifford+$T$ circuits, up to given epsilon. We present a fast new probabilistic algorithm for solving this problem optimally, i.e., for finding the shortest possible circuit whatsoever for the given problem instance. The algorithm requires a factoring oracle (such as a quantum computer). Even in the absence of a factoring oracle, the algorithm is still near-optimal under a mild number-theoretic hypothesis. In this case, the algorithm finds a solution of $T$-count $m + O(\log(\log(1/\epsilon)))$, where $m$ is the $T$-count of the
second-to-optimal solution. In the typical case, this yields circuit approximations of $T$-count $3\log_2(1/\epsilon) + O(\log(\log(1/\epsilon)))$. Our algorithm is efficient in practice, and provably efficient under the above-mentioned number-theoretic hypothesis, in the sense that its expected runtime is $O(\polylog(1/\epsilon))$.

Coherent modification of entanglement: benefits due to extended Hilbert space (pp0954-0968)
          
Dmitry Solenov
A quantum computing system is typically represented by a set of non-interacting (local) two-state systems---qubits. Many physical systems can naturally have more accessible states, both local and non-local. We show that the resulting non-local network of states connecting qubits can be efficiently addressed via continuous time quantum random walks, leading to substantial speed-up of multiqubit entanglement manipulations. We discuss a three-qubit Toffoli gate and a system of superconducting qubits as an illustration.

The entangling power of a glocal dissipative map (pp0969-0981)
          
Alireza Nourmandipour, M.K. Tavassoly, and Stefano Mancini
We consider a model of two qubits dissipating into both local and global environments (generally at non-zero temperatures), with the possibility of interpolating between purely local dissipation and purely global one. The corresponding dissipative dynamical map is characterized in terms of its Kraus operators focusing on the stationary regime. We then determine conditions under which entanglement can be induced by the action of such a map. It results (rather counterintuitively) that in order to have entanglement in the presence of local environment, this latter must be at nonzero temperature.

Environment-assisted entanglement purification (pp0982-0990)
          
Liang Qiu, Zhi Liu, and Xin Wang
Two qubits in a pure entangled state passing through and interacting with amplitude damping noises will cause the decay of entanglement. Entanglement swapping combined with environment measurement is proposed to purify entanglement of the two-qubit state. Some initial states can be purified into the maximally entangled ones by just using the protocol for one time, in contrast to iteratively using the protocol given in Phys. Rev. A 89, 014303 (2014).

Optimal universal quantum cloning: asymmetries and fidelity measures (pp0991-1028)
          
Alastair Kay
We study the problem of universal quantum cloning -- taking several identical copies of a pure but unknown quantum state and producing further copies. While it is well known that it is impossible to perfectly reproduce the state, how well the copies can be cloned can be quantified using the fidelity. We examine how individual fidelities can be traded against each other, and how different fidelity measures can be incorporated. The broadly applicable formalism into which we transform the cloning problem is described as a series of quadratic constraints which are amenable to mathematical and computational scrutiny. As such, we reproduce all known results on optimal universal cloning, and push the recent results on asymmetric cloning much further, giving new trade-off relations between fidelities for broad classes of optimal cloning machines. We also provide substantial evidence that motivates why other parameter ranges (number of input copies) have not, and will not, yield to similar analysis.

Solving constrained quadratic binary problems via quantum adiabatic evolution (pp1029-1047)
          
Pooya Ronagh, Brad Woods, and Ehsan Iranmanesh
Quantum adiabatic evolution is perceived as useful for binary quadratic programming problems that are a priori unconstrained. For constrained problems, it is a common practice to relax linear equality constraints as penalty terms in the objective function. However, there has not yet been proposed a method for efficiently dealing with inequality constraints using the quantum adiabatic approach. In this paper, we give a method for solving the Lagrangian dual of a binary quadratic programming (BQP) problem in the presence of inequality constraints and employ this procedure within a branch-and-bound framework for constrained BQP (CBQP) problems.

An adaptive attack on Wiesner's quantum money (pp1048-1070)
          
Daniel Nagaj, Or Sattath, Aharon Brodutch, and Dominique Unruh
Unlike classical money, which is hard to forge for practical reasons (e.g. producing paper with a certain property), quantum money is attractive because its security might be based on the no-cloning theorem. The first quantum money scheme was introduced by Wiesner circa 1970. Although more sophisticated quantum money schemes were proposed, Wiesner's scheme remained appealing because it is both conceptually clean as well as relatively easy to implement. We show efficient adaptive attacks on Wiesner's quantum money scheme (and its variant by Bennett et al.), when valid money is accepted and passed on, while invalid money is destroyed. We propose two attacks, the first is inspired by the Elitzur-Vaidman bomb testing problem, while the second is based on the idea of {\it protective measurements}. It allows us to break Wiesner's scheme with 4 possible states per qubit, and generalizations which use more than 4 states per qubit. The attack shows that Wiesner's scheme can only be safe if the bank replaces valid notes after validation.

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