Neighbor Architectures Research Articles

An Efficient Method to Decompose and Map MPMCT Gates That Accounts for Qubit Placement

Mixed-Polarity Multiple-Control Toffoli (MPMCT) gates are generally used to implement large control logic functions for quantum computation. A logic circuit consisting of MPMCT gates needs to be mapped to a quantum computing device that invariably has a physical limitation, which means we need to (1) decompose the MPMCT gates into one- or two-qubit gates, and then (2) insert SWAP gates so that all the gates can be performed on Nearest Neighbor Architectures (NNAs). Up to date, the above two processes have only been studied independently. In this work, we investigate that the total number of gates in a circuit can be decreased if the above two processes are considered simultaneously as a single step. We developed a method that inserts SWAP gates while decomposing MPMCT gates unlike most of the existing methods. Also, we consider the effect on the latter part of a circuit carefully by considering the qubit placement when decomposing an MPMCT gate. Experimental results demonstrate the effectiveness of our method.

Heuristic Reordering Strategy for Quantum Circuit Mapping on LNN Architectures.

Because of the connection constraints of quantum devices, the quantum gate cannot operate directly on nonadjacent qubits. Quantum circuit mapping transforms a logical quantum circuit to a circuit that satisfies the connection constraints by adding SWAP gates for nonadjacent qubits. Global and local heuristic reordering strategies are proposed in this paper for quantum circuit mapping over linear nearest neighbor (LNN) architectures, which are one-dimensional topology structures, to reduce the number of SWAP gates added. Experiment results show that the average improvements of the two methods are 13.19% and 15.46%, respectively. In this paper, we consider the quantum circuit mapping problem for linear nearest neighbor (LNN) architectures. We propose a global heuristic qubit reordering optimization algorithm and a local heuristic qubit reordering optimization algorithm. Compared with the other algorithm results, the average improvements of the two methods for quantum cost are 13.19% and 15.46%, respectively. The two methods apply to the realization of quantum circuit neighboring over one-dimensional quantum architectures and can be extended to algorithms that work for other quantum architectures of different topologies.

Hadamard-Free Circuits Expose the Structure of the Clifford Group

The Clifford group plays a central role in quantum randomized benchmarking, quantum tomography, and error correction protocols. Here we study the structural properties of this group. We show that any Clifford operator can be uniquely written in the canonical form F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> HSF <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , where H is a layer of Hadamard gates, S is a permutation of qubits, and F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> are parameterized Hadamard-free circuits chosen from suitable subgroups of the Clifford group. Our canonical form provides a one-to-one correspondence between Clifford operators and layered quantum circuits. We report a polynomial-time algorithm for computing the canonical form. We employ this canonical form to generate a random uniformly distributed n-qubit Clifford operator in runtime O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). The number of random bits consumed by the algorithm matches the information-theoretic lower bound. A surprising connection is highlighted between random uniform Clifford operators and the Mallows distribution on the symmetric group. The variants of the canonical form, one with a short Hadamard-free part and one allowing a circuit depth 9n implementation of arbitrary Clifford unitaries in the Linear Nearest Neighbor architecture are also discussed. Finally, we study computational quantum advantage where a classical reversible linear circuit can be implemented more efficiently using Clifford gates, and show an explicit example where such an advantage takes place.

An ant colony based mapping of quantum circuits to nearest neighbor architectures

Although this decade is witnessing tremendous advancements in fabrication technologies for quantum circuits, this industry is facing several design challenges and technological constraints. Nearest Neighbor (NN) enforcement is one such design constraint that demands the physical qubits to be adjacent. In the last couple of years, this domain has made progress starting from designing advanced algorithms to improved synthesis methodologies, even though developing efficient design solutions remains an active area of research.Here, we propose such a synthesis technique that efficiently transforms quantum circuits to NN designs. To find the NN solution, we have taken help of an ant colony algorithm which completes the circuit conversion in two phases: in the first phase, it finds the global qubit ordering for the input circuit and, in the second phase, a heuristic driven look-ahead scheme is executed for local reordering of gates. The proposed algorithm is first fitted into a 1D design and, later, mapped to 2D and 3D configurations. The combination of such heuristic and the meta-heuristic schemes has resulted promising solutions in the transformation of quantum circuits to NN-compliant architectures. We have tested our algorithm over a wide spectrum of benchmarks and comparisons with state-of-the-art design approaches showed considerable improvements.

Mapping a Quantum Circuit to 2D Nearest Neighbor Architecture by Changing the Gate Order

Mapping a Quantum Circuit to 2D Nearest Neighbor Architecture by Changing the Gate Order

Advantages of a modular high-level quantum programming framework

We review some of the features of the ProjectQ software framework and quantify their impact on the resulting circuits. The concise high-level language facilitates implementing even complex algorithms in a very time-efficient manner while, at the same time, providing the compiler with additional information for optimization through code annotation – so-called meta-instructions. We investigate the impact of these annotations for the example of Shor’s algorithm in terms of logical gate counts. Furthermore, we analyze the effect of different intermediate gate sets for optimization and how the dimensions of the resulting circuit depend on a smart choice thereof. Finally, we demonstrate the benefits of a modular compilation framework by implementing mapping procedures for one- and two-dimensional nearest neighbor architectures which we then compare in terms of overhead for different problem sizes.

Shorter Stabilizer Circuits via Bruhat Decomposition and Quantum Circuit Transformations

In this paper, we improve the layered implementation of arbitrary stabilizer circuits introduced by Aaronson and Gottesman in Phys. Rev. A 70 (052328), 2004: to implement a general stabilizer circuit, we reduce their 11-stage computation -H-C-P-C-P-C-H-P-C-P-Cover the gate set consisting of Hadamard, controlled-NOT, and phase gates, into a 7-stage computation of the form -C-CZ-P-H-P-CZ-C-. We show arguments in support of using -CZ- stages over the -C- stages: not only the use of -CZ- stages allows a shorter layered expression, but -CZ- stages are simpler and appear to be easier to implement compared to the -C- stages. Based on this decomposition, we develop a two-qubit gate depth(14n-4) implementation of stabilizer circuits over the gate library {H, P, CNOT}, executable in the Linear Nearest Neighbor (LNN) architecture, improving best previously known depth25n circuit, also executable in the LNN architecture. Our constructions rely on Bruhat decomposition of the symplectic group and on folding arbitrarily long sequences of the form (-P-C-) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> into a three-stage computation -P-CZ-C-. Our results include the reduction of the 11-stage decomposition -H-C-P-C-PC-H-P-C-P-Cinto a 9-stage decomposition of the form -C-P-C-PH-C-P-C-P-. This reduction is based on the Bruhat decomposition of the symplectic group. This result also implies a new normal form for stabilizer circuits. We show that a circuit in this normal form is optimal in the number of Hadamard gates used. We also show that the normal form has an asymptotically optimal number of parameters.

Quantum State Analysis and Optimal Logic Synthesis for Linear Nearest Neighbor Architecture

Real quantum computing technologies have different restrictions and constraints which need to be considered during circuit synthesis. In certain technologies, only physically adjacent qubits can interact. It is common to use a pair of SWAP gates to bring the control and target qubits of any quantum gate to adjacent lines and maintain correct circuit functionality. This paper introduces a new method to synthesize linear nearest neighbor quantum circuit on the base of the analysis of quantum state in circuit. For all 3-qubit quantum circuit, experimental results shows that our algorithm can reduce the quantum cost by approximately 30% on average compared with one recent proposed method.

Layout Optimization for Quantum Circuits with Linear Nearest Neighbor Architectures

This paper is concerned with the physical design of quantum logic circuits. More precisely, it addresses the problem of minimizing the number of required qubit reorderings (achieved by inserting explicit SWAP gates) when mapping a quantum circuit into a linear nearest neighbor quantum archi-tecture. First, an interaction graph that captures the interaction distances among various qubits in the quantum circuit is constructed. The interaction graph is utilized to partition the quantum circuit into a set of subcircuits such that the number of required qubit reoderings within each subcircuit is provably no more than a given threshold. Next, a Minimum Linear Arrangement problem for each subcircuit is formulated and solved to achieve the minimum number of internal qubit reorderings and determine the subcircuit input and output qubit orderings. Finally, a bubble sort algorithm is repeatedly employed to minimize the number of qubit reorderings that are required between the consecutive subcircuits. Experiments done on various quantum Fourier transform circuits as well as various reversible logic circuits demonstrate the effectiveness of the proposed approach.

Gravitational fixed radius nearest neighbor for imbalanced problem

This paper proposes a novel learning model that introduces the calculation of the pairwise gravitation of the selected patterns into the classical fixed radius nearest neighbor method, in order to overcome the drawback of the original nearest neighbor rule when dealing with imbalanced data. The traditional k nearest neighbor rule is considered to lose power on imbalanced datasets because the final decision might be dominated by the patterns from negative classes in spite of the distance measurements. Differently from the existing modified nearest neighbor learning model, the proposed method named GFRNN has a simple structure and thus becomes easy to work. Moreover, all parameters of GFRNN do not need initializing or coordinating during the whole learning procedure. In practice, GFRNN first selects patterns as candidates out of the training set under the fixed radius nearest neighbor rule, and then introduces the metric based on the modified law of gravitation in the physical world to measure the distance between the query pattern and each candidate. Finally, GFRNN makes the decision based on the sum of all the corresponding gravitational forces from the candidates on the query pattern. The experimental comparison validates both the effectiveness and the efficiency of GFRNN on forty imbalanced datasets, comparing to nine typical methods. As a conclusion, the contribution of this paper is constructing a new simple nearest neighbor architecture to deal with imbalanced classification effectively without any manually parameter coordination, and further expanding the family of the nearest neighbor based rules.

Line ordering of reversible circuits for linear nearest neighbor realization

Real quantum computing technologies have different restrictions and constraints which need to be considered during circuit synthesis. In certain technologies, only physically adjacent qubits can interact, which restricts their realizations to only linear nearest neighbor (LNN) architecture. In this work, we formulate the line ordering problem in LNN architecture as task assignment problem to find a mapping (permutation) between task graph and processor graph with minimum cost. We propose two different approaches, a greedy heuristic and a meta-heuristic algorithm based on Harmony Search to solve the task assignment problem. Experimental results show that our algorithms were able to reduce the quantum cost of benchmark circuits by approximately 30 % on average. Moreover, the proposed algorithms were compared to one recently proposed ordering algorithm and were found to further improve the cost by approximately 16 %.

Efficient quantum computing between remote qubits in linear nearest neighbor architectures

We propose a new scheme for implementing gate operations between remote qubits in linear nearest neighbor (LNN) architectures, one that does not require qubits to be adjacent to each other in order...

Depth-optimized reversible circuit synthesis

In this paper, simultaneous reduction of circuit depth and synthesis cost of reversible circuits in quantum technologies with limited interaction is addressed. We developed a cycle-based synthesis algorithm which uses negative controls and limited distance between gate lines. To improve circuit depth, a new parallel structure is introduced in which before synthesis a set of disjoint cycles are extracted from the input specification and distributed into some subsets. The cycles of each subset are synthesized independently on different sets of ancillae. Accordingly, each disjoint set can be synthesized by different synthesis methods. Our analysis shows that the best worst-case synthesis cost of reversible circuits in the linear nearest neighbor architecture is improved by the proposed approach. Our experimental results reveal the effectiveness of the proposed approach to reduce cost and circuit depth for several benchmarks.

Towards Practical Classical Processing for the Surface Code

The surface code is unarguably the leading quantum error correction code for 2D nearest neighbor architectures, featuring a high threshold error rate of approximately 1%, low overhead implementations of the entire Clifford group, and flexible, arbitrarily long-range logical gates. These highly desirable features come at the cost of significant classical processing complexity. We show how to perform the processing associated with an n×n lattice of qubits, each being manipulated in a realistic, fault-tolerant manner, in O(n2) average time per round of error correction. We also describe how to parallelize the algorithm to achieve O(1) average processing per round, using only constant computing resources per unit area and local communication. Both of these complexities are optimal.

A nearest neighbor architecture to overcome dephasing

We design a nearest-neighbor architectural layout that uses fixed positive and negative couplings between qubits, to overcome the effects of relative phases due to qubit precessions, both during idle times and gate operations. The scheme uses decoherence-free subspaces, and we show how to realize gate operations on these encoded qubits. The main advantage of our scheme is that most gate operations are realized by only varying a single control parameter, which greatly reduces the circuit complexity. Moreover, the scheme is robust against phase errors occurring as a result of finite rise and fall times due to non-ideal pulses.

Universal quantum computing in linear nearest neighbor architectures

We introduce a scheme for realizing universal quantum computing in a linear nearest neighbor architecture with fixed couplings. We first show how to realize a controlled-NOT gate operation between two adjacent qubits without having to isolate the two qubits from qubits adjacent to them. The gate operation is implemented by applying two consecutive pulses of equal duration, but varying amplitudes, on the target qubit. Since only a single control parameter is required in implementing our scheme, it is very efficient. We next show how our scheme can be used to realize single qubit rotations and two-qubit controlled-unitary operations. As most proposals for solid state implementations of a quantum computer use a one-dimensional line of qubits, the schemes presented here will be extremely useful.

An efficient conversion of quantum circuits to a linear nearest neighbor architecture

Several promising implementations of quantum computation rely on a Linear Nearest Neighbor (LNN) architecture, which arranges quantum bits on a line, and allows neighbor interactions only. Therefore, several specific circuits have been designed on an LNN architecture. However, a general and efficient conversion method for an arbitrary circuit has not been established. Therefore, this paper gives an efficient conversion technique to convert quantum circuits to an LNN architecture. When a quantum circuit is converted to an LNN architecture, the objective is to reduce the size of the additional circuit added by the conversion and the time complexity of the conversion. The proposed method requires less additional circuitry and time complexity compared with naive techniques. To develop the method, we introduce two key theorems that may be interesting on their own. In addition, the proposed method also achieves less overhead than some known circuits designed from scratch on an LNN architecture.

Synthesis of quantum circuits in Linear Nearest neighbor Model using Positive Davio Lattices

We present a logic synthesis method based on lattices that realize quantum arrays in One-Dimensional Ion Trap technology. This means that all gates are built from 2x2 quantum primitives that are located only on neighbor qubits in a one-dimensional space (called also vector of qubits or Linear Nearest Neighbor (LNN) architecture). The Logic circuits designed by the proposed method are realized only with 3*3 Toffoli, Feynman and NOT quantum gates and the usage of the commonly used multi-input Toffoli gates is avoided. This realization method of quantum circuits is different from most of reversible circuits synthesis methods from the literature that use only high level quantum cost based on the number of quantum gates. Our synthesis approach applies to both standard and LNN quantum cost models. It leads to entirely new CAD algorithms for circuit synthesis and substantially decreases the quantum cost for LNN quantum circuits. The drawback of synthesizing circuits in the presented LNN architecture is the addition of ancilla qubits.

Synthesis of quantum circuits for linear nearest neighbor architectures

While a couple of impressive quantum technologies have been proposed, they have several intrinsic limitations which must be considered by circuit designers to produce realizable circuits. Limited interaction distance between gate qubits is one of the most common limitations. In this paper, we suggest extensions of the existing synthesis flow aimed to realize circuits for quantum architectures with linear nearest neighbor (LNN) interaction. To this end, a template matching optimization, an exact synthesis approach, and two reordering strategies are introduced. The proposed methods are combined as an integrated synthesis flow. Experiments show that by using the suggested flow, quantum cost can be improved by more than 50% on average.

An O(m 2 )-depth quantum algorithm for the elliptic curve discrete logarithm problem over GF(2 m ) a

We consider a quantum polynomial-time algorithm which solves the discrete logarithmproblem for points on elliptic curves over GF(2m). We improve over earlier algorithmsby constructing an efficient circuit for multiplying elements of binary finite fields andby representing elliptic curve points using a technique based on projective coordinates.The depth of our proposed implementation, executable in the Linear Nearest Neighbor(LNN) architecture, is O(m2), which is an improvement over the previous bound ofO(m3) derived assuming no architectural restrictions.