Operator-Schmidt decomposition and the geometrical edges of two-qubit gates

Nonlocal two-qubit quantum gates are represented by canonical decomposition or equivalently by operator-Schmidt decomposition. The former decomposition results in geometrical representation such that all the two-qubit gates form tetrahedron within which perfect entanglers form a polyhedron. On the other hand, it is known from the later decomposition that Schmidt number of nonlocal gates can be either 2 or 4. In this work, some aspects of later decomposition are investigated. It is shown that two gates differing by local operations possess same set of Schmidt coefficients. Employing geometrical method, it is established that Schmidt number 2 corresponds to controlled unitary gates. Further, all the edges of tetrahedron and polyhedron are characterized using Schmidt strength, a measure of operator entanglement. It is found that one edge of the tetrahedron possesses the maximum Schmidt strength, implying that all the gates in the edge are maximally entangled.     

Multi-party quantum state sharing via various probabilistic channels

We present a new scheme to share an arbitrary multi-qubit state between n agents via various probabilistic channels under cooperation of m?1 controllers with a certain probability. Compared with existing ones in this literature, our scheme involves various probabilistic channels, which weakens the requirement for quantum channels. The proposed scheme is symmetric which means even though the designed receiver has no capability of adopting appropriate strategies in introducing auxiliary qubits and performing two-qubit gates, it is still possible to faithfully share a multi-qubit state with assistance of other participants. This scheme involves only single-qubit measurements, CNOT gates, and local two-qubit gates with an auxiliary qubit, which makes it more convenient for physical realization.     

An efficient quantum circuit analyser on qubits and qudits

This paper presents a highly efficient decomposition scheme and its associated Mathematica notebook for the analysis of complicated quantum circuits comprised of single/multiple qubit and qudit quantum gates. In particular, this scheme reduces the evaluation of multiple unitary gate operations with many conditionals to just two matrix additions, regardless of the number of conditionals or gate dimensions. This improves significantly the capability of a quantum circuit analyser implemented in a classical computer. This is also the first efficient quantum circuit analyser to include qudit quantum logic gates.

Program summary

Program title:CUGates.mCatalogue identifier: AEJM_v1_0Program summary: URL: http://cpc.cs.qub.ac.uk/summaries/AEJM_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 8168No. of bytes in distributed program, including test data, etc.: 173 899Distribution format: tar.gzProgramming language: MathematicaComputer: Any computer installed with Mathematica 6.0 or higher.Operating system: Any system with a copy of Mathematica 6.0 or higher installed.Classification: 4.15Nature of problem: The CUGates notebook simulates arbitrarily complex quantum circuits comprised of single/multiple qubit and qudit quantum gates.Solution method: It utilizes an irreducible form of matrix decomposition for a general controlled gate with multiple conditionals and is highly efficient in simulating complex quantum circuits.Running time: Details of CPU time usage for various example runs are given in Section 4.     

Construction of four-qubit quantum entanglement for SI (S = 3/2, I = 3/2) spin system

In quantum information processing, spin-3/2 electron or nuclear spin states are known as two-qubit states. For SI (S = 3/2, I = 3/2) spin system, there are 16 four-qubit states. In this study, first, four-qubit entangled states are obtained by using the matrix representation of Hadamard and CNOT logic gates. By considering ⁷⁵As@C₆₀ molecule as SI (S = 3/2, I = 3/2) spin system, four-qubit entangled states are also obtained by using the magnetic resonance pulse sequences of Hadamard and CNOT logic gates. Then, it is shown that obtained entangled states can be transformed into each other by the transformation operators.     

Circuit QED: implementation of the three-qubit refined Deutsch–Jozsa quantum algorithm

We propose a protocol to construct the 35 \(f\) -controlled phase gates of a three-qubit refined Deutsch–Jozsa (DJ) algorithm, by using single-qubit \(\sigma _z\) gates, two-qubit controlled phase gates, and two-target-qubit controlled phase gates. Using this protocol, we discuss how to implement the three-qubit refined DJ algorithm with superconducting transmon qutrits resonantly coupled to a single cavity. Our numerical calculation shows that implementation of this quantum algorithm is feasible within the present circuit QED technique. The experimental realization of this algorithm would be an important step toward more complex quantum computation in circuit QED.     

Towards Scalable Linear-Optical Quantum Computers

Scalable quantum computation with linear optics was considered to be impossible due to the lack of efficient two-qubit logic gates, despite the ease of implementation of one-qubit gates. Two-qubit gates necessarily need a non-linear interaction between the two photons, and the efficiency of this non-linear interaction is typically very small in bulk materials. However, it has recently been shown that this barrier can be circumvented with effective non-linearities produced by projective measurements, and with this work linear-optical quantum computing becomes a new avenue towards scalable quantum computation. We review several issues concerning the principles and requirements of this scheme. PACS: 03.67.Lx, 03.67.Pp, 42.50.Dv, 42.65.Lm     

量子搜索算法体系的仿真实现及研究

核磁共振（NMR）技术被认为是最为有效的实现量子计算的物理体系之一。多量子算符代数理论可以将幺正变换分解为一系列有限的单量子门和对角双量子门的组合。本文以核磁共振和多量子算符代数理论为基础，提出了实现任意相位旋转角度的一般化量子搜索算法的核磁共振脉冲序列设计方法，并在量子计算仿真程序上进行了双量子位的不同相位旋转角度的量子搜索算法的实验验证。     

Controlled gates for multi-level quantum computation

Multi-level (ML) quantum logic can potentially reduce the number of inputs/outputs or quantum cells in a quantum circuit which is a limitation in current quantum technology. In this paper we propose theorems about ML-quantum and reversible logic circuits. New efficient implementations for some basic controlled ML-quantum logic gates, such as three-qudit controlled NOT, Cycle, and Self Shift gates are proposed. We also propose lemmas about r-level quantum arrays and the number of required gates for an arbitrary n-qudit ML gate. An equivalent definition of quantum cost (QC) of binary quantum gates for ML-quantum gates is introduced and QC of controlled quantum gates is calculated.     

Constructing quantum logic gates using q-deformed harmonic oscillator algebras

We study two-level q-deformed angular momentum states, and using q-deformed harmonic oscillators, we provide a framework for constructing qubits and quantum gates. We also present the construction of some basic one-qubit and two-qubit quantum logic gates.     

On the role of the four-qubit state in two-qubit gate teleportation

The full analysis of quantum protocols requires the knowledge of the role of quantum states, bases of measurement and quantum gates involved. In what concerns the famous two-qubit quantum gate teleportation protocol, the role of the basis of measurement was considered in a recent work by Mendes and Ramos. In this work, we analyze the role of the four-qubit state used as resource. We show that the quantum two-qubit gate teleportation divides the set of pure four-qubit states in two classes. For one class, deterministic and probabilistic teleportation can be achieved, while for the other class, probabilistic remote two-qubit gate preparation is achieved.     

Direct implementation of an N-qubit controlled-unitary gate in a single step

We present a new scheme to implement an N-qubit controlled-unitary operation directly in a single step. The main advantage of our scheme is that we do not use conventional gate decomposition protocols to break an N-qubit controlled-unitary gate into one- and two-qubit gates. This greatly reduces the number of computational steps in implementing quantum algorithms and error-correcting codes, which use multi-control unitary operations. We show how to find analytic solutions to the time evolution of the system, so that system parameters can be found to realize the desired N-qubit controlled-unitary operations.     

Mathematical models of quantum computation

In this paper, we introduce two mathematical models of realistic quantum computation. First, we develop a theory of bulk quantum computation such as NMR (Nuclear Magnetic Resonance) quantum computation. For this purpose, we define bulk quantum Turing machine (BQTM for short) as a model of bulk quantum computation. Then, we define complexity classes EBQP, BBQP and ZBQP as counterparts of the quantum complexity classes EQP, BQP and ZQP, respectively, and show that EBQP=EQP, BBQP=BQP and ZBQP=ZQP. This implies that BQTMs are polynomially related to ordinary QTMs as long as they are used to solve decision problems. We also show that these two types of QTMs are also polynomially related when they solve a function problem which has a unique solution. Furthermore, we show that BQTMs can solve certain instances of NP-complete problems efficiently. On the other hand, in the theory of quantum computation, only feed-forward quantum circuits are investigated, because a quantum circuit represents a sequence of applications of time evolution operators. But, if a quantum computer is a physical device where the gates are interactions controlled by a current computer such as laser pulses on trapped ions, NMR and most implementation proposals, it is natural to describe quantum circuits as ones that have feedback loops if we want to visualize the total amount of the necessary hardware. For this purpose, we introduce a quantum recurrent circuit model, which is a quantum circuit with feedback loops. LetC be a quantum recurrent circuit which solves the satisfiability problem for a blackbox Boolean function includingn variables with probability at least 1/2. And lets be the size ofC (i.e. the number of the gates inC) andt be the number of iterations that is needed forC to solve the satisfiability problem. Then, we show that, for those quantum recurrent circuits, the minimum value ofmax(s, t) isO(n ²2 ^n/3). Tetsuro Nishino, D.Sc.: He is presently an Associate Professor in the Department of Information and Communication Engineering, The University of Electro-Communications. He received the B.S., M.S. and D.Sc degrees in mathematics from Waseda University, in 1982, 1984 and 1991 respectively. From 1984 to 1987, he joined Tokyo Research Laboratory, IBM Japan. From 1987 to 1992, he was a Research Associate of Tokyo Denki University, and from 1992 to 1994, he was an Associate Professor of Japan Advanced Institute of Science and Technology, Hokuriku. His main interests are circuit complexity theory, computational learning theory and quantum complexity theory.     

Efficient protocol of N-bit discrete quantum Fourier transform via transmon qubits coupled to a resonator

Based on the one- and two-qubit gates defined and generated via superconducting transmon qubits homogeneously coupled to a superconducting stripline resonator, we present a new physical protocol for implementing an $N$ -bit discrete quantum Fourier transform. We propose and illustrate a detailed experimental feasibility for realizing the algorithm. The average fidelity is computed to prove the success of this algorithm. Estimated time for implementing the protocol using the proposed scheme is compared with previous schemes. Estimates show that the protocol can be successfully implemented within the present experimental limits.     

Quantum computation in the decoherence-free subspaces with cavity QED

We present a scheme to implement quantum computation in decoherence-free subspaces (DFSs) with four atoms in a single-mode cavity. A four-dimensional DFS is constituted to protect quantum information when the full symmetry of interaction between system and environment is broken in a specific way, and entangling two-qubit logic gates and noncommuting single-qubit gates are implemented in such DFS. The gate fidelity is numerically calculated, and the feasibility of the approximations taken in this work is verified based on the numerical calculations.     

多量子位Grover量子搜索算法的NMR仿真实现

核磁共振（NMR）技术目前是能有效实现量子计算的物理体系之一。多量子算符代数理论可以将幺正变换分解为一系列有限的单量子门和对角双量子门的组合。本文以核磁共振和多量子算符代数理论为基础,提出了实现多量子位Grover量子搜索算法的核磁共振脉冲序列设计方法,并在量子计算仿真程序上进行了3量子位的Grover量子搜索算法的实验验证。     

On figures of merit in reversible and quantum logic designs

Five figures of merit including number of gates, quantum cost, number of constant inputs, number of garbage outputs, and delay are used casually in the literature to compare the performance of different reversible or quantum logic circuits. In this paper we propose new definitions and enhancements, and identify similarities between these figures of merit. We evaluate these measures to show their strength and weakness. Instead of the number of gates, we introduce the weighted number of gates, where a weighting factor is assigned to each quantum or reversible gate, based on its type, size and technology. We compare the quantum cost with weighted number of gates of a circuit and show three major differences between these measures. It is proved that it is not possible to define a universal reversible logic gate without adding constant inputs. We prove that there is an optimum value for number of constant inputs to obtain a circuit with minimum quantum cost. Some reversible logic benchmarks have been synthesized using Toffoli and Fredkin gates to obtain their optimum values of number of constant inputs. We show that the garbage outputs can also be used to decrease the quantum cost of the circuit. A new definition of delay in quantum and reversible logic circuits is proposed for music line style representation. We also propose a procedure to calculate the delay of a circuit, based on the quantum cost and the depth of the circuit. The results of this research show that to achieve a fair comparison among designs, figures of merit should be considered more thoroughly.      

Scalable quantum computing model in the circuit-QED lattice with circulator function

We propose a model for a scalable quantum computing in the circuit quantum electrodynamics architecture. In the Kagome lattice of qubits, three qubits are connected to each other through a superconducting three-junction flux qubit at the vertices of the lattice. By controlling one of the three-Josephson-junction energies of the intervening flux qubit, we can achieve the circulator function that couples arbitrary pair of two qubits among three. This selective coupling enables the interaction between two nearest neighbor qubits in the Kagome lattice, and further the two-qubit gate operation between any pair of qubits in the whole lattice by performing consecutive nearest neighbor two-qubit gates.     

Nondestructive Greenberger-Horne-Zeilinger-state analyzer

We propose a method to construct a nondestructive n-qubit Greenberger– Horne–Zeilinger (GHZ)-state analyzer. The method is applied to any systems in which two-qubit parity gates, controlled-phase gates, or controlled-NOT gates can be realized. We also present a simplified two-photon parity gate with which a nondestructive n-photon GHZ-state analyzer could be largely simplified. The nondestructive GHZ-state analyzer is expected to find useful applications for economical quantum-information processing.