QIC Abstracts

 Vol.12 No.11&12, November 1, 2012

Research Articles:

Hamiltonian simulation using linear combinations of unitary operations (pp0901-0924)
Andrew M. Childs and Nathan Wiebe

We present a new approach to simulating Hamiltonian dynamics based on implementing linear combinations of unitary operations rather than products of unitary operations. The resulting algorithm has superior performance to existing simulation algorithms based on product formulas and, most notably, scales better with the simulation error than any known Hamiltonian simulation technique. Our main tool is a general method to nearly deterministically implement linear combinations of nearby unitary operations, which we show is optimal among a large class of methods.

Classical simulation of dissipative fermionic linear optics (pp0925-0943)
Sergey Bravyi and Robert Konig
Fermionic linear optics is a limited form of quantum computation which is known to be efficiently simulable on a classical computer. We revisit and extend this result by enlarging the set of available computational gates: in addition to unitaries and measurements, we allow dissipative evolution governed by a Markovian master equation with linear Lindblad operators. We show that this more general form of fermionic computation is also simulable efficiently by classical means. Given a system of $N$~fermionic modes, our algorithm simulates any such gate in time $O(N^3)$ while a single-mode measurement is simulated in time $O(N^2)$. The steady state of the Lindblad equation can be computed in time $O(N^3)$.

Limitation for linear maps in a class for detection and quantification of bipartite nonclassical correlation (pp0944-0952)
Akira SaiToh, Roabeh Rahimi, and Mikio Nakahara
Eigenvalue-preserving-but-not-completely-eigenvalue-preserving (EnCE) maps were previously introduced for the purpose of detection and quantification of nonclassical correlation, employing the
paradigm where nonvanishing quantum discord implies the existence of nonclassical correlation. It is known that only the matrix transposition is nontrivial among Hermiticity-preserving (HP) linear EnCE maps when we use the changes in the eigenvalues of a density matrix due to a partial map for the purpose. In this paper, we prove that this is true even among not-necessarily HP (nnHP) linear EnCE maps. The proof utilizes a conventional theorem on linear preservers. This result imposes a strong limitation on the linear maps and promotes the importance of nonlinear maps.

Large-scale multipartite entanglement in the quantum optical frequency comb (pp0953-0969)
Reihaneh Shahrokhshahi and Olivier Pfister
We show theoretically that multipartite entanglement is generated on a massive scale in the spectrum, or optical frequency comb, of a single optical parametric oscillator (OPO) emitting well above threshold. In this system, the quantum dynamics of the strongly depleted pump field are responsible for the onset of the entanglement by correlating the two-mode squeezed, bipartite-entangled pairs of OPO signal fields. (Such pairs are independent of one another in the undepleted, classical pump approximation.) We verify the multipartite nature of the entanglement by evaluating the van Loock-Furusawa criterion for a particular set of entanglement witnesses deduced from physical considerations.

Low-overhead surface code logical Hadamard (pp0970-0982)
Austin G. Fowler
We present an improved low-overhead implementation of surface code logical Hadamard ($H$). We describe in full detail logical $H$ applied to a single distance-7 double-defect logical qubit in an otherwise idle scalable array of such qubits. Our goal is to provide a clear description of logical $H$ and to emphasize that the surface code possesses low-overhead implementations of the entire Clifford group.

Improved bounds on negativity of superpositions (pp0983-0988)
Zhi-Hao Ma, Zhi-Hua Chen, Shuai Han, Shao-Ming Fei, and Simone Severini
We consider an alternative formula for the negativity based on a simple generalization of the concurrence. We use the formula to bound the amount of entanglement in a superposition of two bipartite pure states of arbitrary dimension. Various examples indicate that our bounds are tighter than the previously known results.

Dirac four-potential tunings-based quantum transistor utilizing the Lorentz force (pp0989-1010)
Agung Trisetyarso

We propose a mathematical model of \textit{quantum} transistor in which bandgap engineering corresponds to the tuning of Dirac potential in the complex four-vector form. The transistor consists of $n$-relativistic spin qubits moving in \textit{classical} external electromagnetic fields. It is shown that the tuning of the direction of the external electromagnetic fields generates perturbation on the potential temporally and spatially, determining the type of quantum logic gates. The theory underlying of this scheme is on the proposal of the intertwining operator for Darboux transfomations on one-dimensional Dirac equation amalgamating the \textit{vector-quantum gates duality} of Pauli matrices. Simultaneous transformation of qubit and energy can be accomplished by setting the $\{\textit{control, cyclic}\}$-operators attached on the coupling between one-qubit quantum gate: the chose of \textit{cyclic}-operator swaps the qubit and energy simultaneously, while \textit{control}-operator ensures the energy conservation.

Finite geometry behind the Harvey-Chryssanthacopoulos four-qubit magic rectangle (pp1011-1016)
Metod Saniga and Michel Planat

A ``magic rectangle" of eleven observables of four qubits, employed by Harvey and Chryssanthacopoulos (2008) to prove the Bell-Kochen-Specker theorem in a 16-dimensional Hilbert space, is given a neat finite-geometrical reinterpretation in terms of the structure of the symplectic polar space $W(7,2)$ of the real four-qubit Pauli group. Each of the four sets of observables of cardinality five represents an elliptic quadric in the three-dimensional projective space of order two (PG$(3,2)$) it spans, whereas the remaining set of cardinality four corresponds to an affine plane of order two. The four ambient PG$(3, 2)$s of the quadrics intersect pairwise in a line, the resulting six lines meeting in a point. Projecting the whole configuration from this distinguished point (observable) one gets another, complementary ``magic rectangle" of the same qualitative structure.

Multicopy programmable discriminators between two unknown qudit states with group-theoretic approach (pp1017-1033)
Tao Zhou, Jing Xin Cui, Xiaohua Wu, and Gui Lu Long
The discrimination between two unknown states can be performed by a universal programmable discriminator, where the copies of the two possible states are stored in two program systems respectively and the copies of data, which we want to confirm, are provided in the data system. In the present paper, we propose a group-theretic approach to the multi-copy programmable state discrimination problem. By equivalence of unknown pure states to known mixed states and with the representation theory of $U(n)$ group, we construct the Jordan basis to derive the analytical results for both the optimal unambiguous discrimination and minimum-error discrimination. The POVM operators for unambiguous discrimination and orthogonal measurement operators for minimum-error discrimination are obtained. We find that the optimal failure probability and minimum-error probability for the discrimination between the mean input mixd states are dependent on the dimension of the unknown qudit states. We applied the approach to generalize the results of He and Bergou (2007) from qubit to qudit case, and we further solve the problem of programmable dicriminators with arbitrary copies of unknown states in both program and data systems.

Fault-tolerant ancilla preparation and noise threshold lower bounds for the 23-qubit Golay code (pp1034-1080)
Adam Paetznick and Ben W. Reichardt

In fault-tolerant quantum computing schemes, the overhead is often dominated by the cost of preparing codewords reliably. This cost generally increases quadratically with the block size of the underlying quantum error-correcting code. In consequence, large codes that are otherwise very efficient have found limited fault-tolerance applications. Fault-tolerant preparation circuits therefore are an important target for optimization.     We study the Golay code, a $23$-qubit quantum error-correcting code that protects the logical qubit to a distance of seven. In simulations, even using a na{\"i}ve ancilla preparation procedure, the Golay code is competitive with other codes both in terms of overhead and the tolerable noise threshold. We provide two simplified circuits for fault-tolerant preparation of Golay code-encoded ancillas. The new circuits minimize error propagation, reducing the overhead by roughly a factor of four compared to standard encoding circuits. By adapting the malignant set counting technique to depolarizing noise, we further prove a threshold above $\threshOverlap$ noise per gate.

back to QIC online Front page