New quantum codes from dual-containing cyclic codes over finite rings

Let \(R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}\), where \(\mathbb {F}_{2^{m}}\) is the finite field with \(2^{m}\) elements, m is a positive integer, and u is an indeterminate with \(u^{k+1}=0.\) In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over R. A new Gray map over R is defined, and a sufficient and necessary condition for the existence of dual-containing cyclic codes over R is given. A new family of \(2^{m}\)-ary quantum codes is obtained via the Gray map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R. In particular, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank–Shor–Steane construction from dual-containing cyclic codes over R.     

Survey of control performance in quantum information processing

There is a rich variety of physics underlying the fundamental gating operations for quantum information processing (QIP). A key aspect of a QIP system is how noise may enter during quantum operations and how suppressing or correcting its effects can best be addressed. Quantum control techniques have been developed to specifically address this effort, although a detailed classification of the compatibility of controls schemes with noise sources found in common quantum systems has not yet been performed. This work numerically examines the performance of modern control methods for suppressing decoherence in the presence of noise forms found in viable quantum systems. The noise-averaged process matrix for controlled one-qubit and two-qubit operations are calculated across noise found in systems driven by Markovian open quantum dynamics. Rather than aiming to describe the absolute best control scheme for a given physical circumstance, this work serves instead to classify quantum control behavior across a large class of noise forms so that opportunities for improving QIP performance may be identified.     

Propagation of correlations in local random quantum circuits

We derive a dynamical bound on the propagation of correlations in local random quantum circuits—lattice spin systems where piecewise quantum operations—in space and time—occur with classical probabilities. Correlations are quantified by the Frobenius norm of the commutator of two positive operators acting on disjoint regions of a one-dimensional circular chain of length L. For a time \(t=O(L)\), correlations spread ballistically to spatial distances \(\mathcal {D}=t\), growing at best, diffusively with time for any distance within that radius with extensively suppressed distance- dependent corrections. For \(t=\varOmega (L^2)\), all parts of the system get almost equally correlated with exponentially suppressed distance- dependent corrections and approach the maximum amount of correlations that may be established asymptotically.     

Constructive quantum scaling of unitary matrices

In this work, we present a method of decomposition of arbitrary unitary matrix \(U\in \mathbf {U}(2^k)\) into a product of single-qubit negator and controlled-\(\sqrt{\text{ NOT }}\) gates. Since the product results with negator matrix, which can be treated as complex analogue of bistochastic matrix, our method can be seen as complex analogue of Sinkhorn–Knopp algorithm, where diagonal matrices are replaced by adding and removing an one-qubit ancilla. The decomposition can be found constructively, and resulting circuit consists of \(O(4^k)\) entangling gates, which is proved to be optimal. An example of such transformation is presented.     

Central limit theorem for reducible and irreducible open quantum walks

In this work we aim at proving central limit theorems for open quantum walks on \({\mathbb {Z}}^d\). We study the case when there are various classes of vertices in the network. In particular, we investigate two ways of distributing the vertex classes in the network. First, we assign the classes in a regular pattern. Secondly, we assign each vertex a random class with a transition invariant distribution. For each way of distributing vertex classes, we obtain an appropriate central limit theorem, illustrated by numerical examples. These theorems may have application in the study of complex systems in quantum biology and dissipative quantum computation.     

Analytical error analysis of Clifford gates by the fault-path tracer method

We estimate the success probability of quantum protocols composed of Clifford operations in the presence of Pauli errors. Our method is derived from the fault-point formalism previously used to determine the success rate of low-distance error correction codes. Here we apply it to a wider range of quantum protocols and identify circuit structures that allow for efficient calculation of the exact success probability and even the final distribution of output states. As examples, we apply our method to the Bernstein–Vazirani algorithm and the Steane [[7,1,3]] quantum error correction code and compare the results to Monte Carlo simulations.     

Study on the security of the authentication scheme with key recycling in QKD

In quantum key distribution (QKD), the information theoretically secure authentication is necessary to guarantee the integrity and authenticity of the exchanged information over the classical channel. In order to reduce the key consumption, the authentication scheme with key recycling (KR), in which a secret but fixed hash function is used for multiple messages while each tag is encrypted with a one-time pad (OTP), is preferred in QKD. Based on the assumption that the OTP key is perfect, the security of the authentication scheme has be proved. However, the OTP key of authentication in a practical QKD system is not perfect. How the imperfect OTP affects the security of authentication scheme with KR is analyzed thoroughly in this paper. In a practical QKD, the information of the OTP key resulting from QKD is partially leaked to the adversary. Although the information leakage is usually so little to be neglected, it will lead to the increasing degraded security of the authentication scheme as the system runs continuously. Both our theoretical analysis and simulation results demonstrate that the security level of authentication scheme with KR, mainly indicated by its substitution probability, degrades exponentially in the number of rounds and gradually diminishes to zero.     

Quantum state transfer between an optomechanical cavity and a diamond nuclear spin ensemble

We explore an efficient scheme for transferring quantum state between an optomechanical cavity and nuclear spins of nitrogen-vacancy centers in diamond, where quantum information can be efficiently stored (retrieved) into (from) the nuclear spin ensemble assisted by a mechanical resonator in a dispersive regime. Our scheme works for a broad range of cavity frequencies and might have potential applications in employing the nuclear spin ensemble as a memory in quantum information processing. The feasibility of our protocol is analyzed using currently available parameters.     

Unitary equivalent classes of one-dimensional quantum walks

This study investigates unitary equivalent classes of one-dimensional quantum walks. We prove that one-dimensional quantum walks are unitary equivalent to quantum walks of Ambainis type and that translation-invariant one-dimensional quantum walks are Szegedy walks. We also present a necessary and sufficient condition for a one-dimensional quantum walk to be a Szegedy walk.