Synthesis Of Quantum Circuits Research Articles

Quantum arithmetic and numerical analysis using repeat-until-success circuits

We develop a method for approximate synthesis of single-qubit rotations of the form e-if(ϕ1,...,ϕk)X that is based on the Repeat-Until-Success (RUS) framework for quantum circuit synthesis. We demo...

Quantum arithmetic and numerical analysis using Repeat-Until-Success circuits

We develop a method for approximate synthesis of single-qubit rotations of the form e−if(φ1,...,φk)X that is based on the Repeat-Until-Success (RUS) framework for quantum circuit synthesis. We demonstrate how smooth computable functions f can be synthesized from two basic primitives. This synthesis approach constitutes a manifestly quantum form of arithmetic that differs greatly from the approaches commonly used in quantum algorithms. The key advantage of our approach is that it requires far fewer qubits than existing approaches: as a case in point, we show that using as few as 3 ancilla qubits, one can obtain RUS circuits for approximate multiplication and reciprocals. We also analyze the costs of performing multiplication and inversion on a quantum computer using conventional approaches and find that they can require too many qubits to execute on a small quantum computer, unlike our approach.

Quantum-Logic Synthesis of Hermitian Gates

In this article, the problem of synthesizing a general Hermitian quantum gate into a set of primary quantum gates is addressed. To this end, an extended version of the Jacobi approach for calculating the eigenvalues of Hermitian matrices in linear algebra is considered as the basis of the proposed synthesis method. The quantum circuit synthesis method derived from the Jacobi approach and its optimization challenges are described. It is shown that the proposed method results in multiple-control rotation gates around the y axis, multiple-control phase shift gates, multiple-control NOT gates, and a middle diagonal Hermitian matrix, which can be synthesized to multiple-control Pauli Z gates. Using the proposed approach, it is shown how multiple-control U gates, where U is a single-qubit Hermitian quantum gate, can be implemented using a linear number of elementary gates in terms of circuit lines with the aid of one auxiliary qubit in an arbitrary state.

Optimal synthesis of multivalued quantum circuits

Although many of works have been done in multivalued quantum logic synthesis, the question whether multivalued quantum circuits are more efficient than the conventional binary quantum circuits is still open. In this article we devote to the optimization of generic multivalued quantum circuits. The multivalued quantum Shannon decompositions (QSD) are improved so that the circuits obtained are asymptotically optimal for all dimensionality d. The syntheses of uniformly multifold controlled $R_y$ rotations are also optimized to make the circuits further simplified. Moreover, the theoretical lower bound of complexity for multivalued quantum circuits is investigated, and a quantity known as efficiency index is proposed to evaluate the efficiency of synthesis of various quantum circuits. The algorithm for qudit circuits given here is an efficient synthesis routine which produces best known results for all dimensionality d, and for both cases the number of qudit n is small and that is asymptotic. The multivalued quantum circuits are indeed more efficient than the binary quantum circuits. The facts, the leading factor of the lower bound of complexity for qudit circuits is small by a factor of d-1 in comparison to that for qubit circuits and the asymptotic efficiency index is increased with the increase of dimensionality d, reveal the potential advantage of qudit circuits over generic qubit circuits. The generic n-qudit circuits with $d\geq5$ and generic two-ququart circuits synthesized by the algorithm given here are practical circuits which are more efficient than the most efficient qubit circuits.

PAQCS: Physical Design-Aware Fault-Tolerant Quantum Circuit Synthesis

Quantum circuits consist of a cascade of quantum gates. In a physical design-unaware quantum logic circuit, a gate is assumed to operate on an arbitrary set of quantum bits (qubits), without considering the physical location of the qubits. However, in reality, physical qubits have to be placed on a grid. Each node of the grid represents a qubit. The grid implements the architecture of the quantum computer. A physical constraint often imposed is that quantum gates can only operate on adjacent qubits on the grid. Hence, a communication channel needs to be built if the qubits in the logical circuit are not adjacent. In this paper, we introduce a tool called the physical design-aware fault-tolerant quantum circuit synthesis (PAQCS). It contains two algorithms: one for physical qubit placement and another for routing of communications. With the help of these two algorithms, the overhead of converting a logical to a physical circuit is reduced by 30.1%, on an average, relative to previous work. The optimization algorithms in PAQCS are evaluated on circuits implemented using quantum operations supported by two different quantum physical machine descriptions and three quantum error-correcting codes. They reduce the number of primitive operations by 11.5%–68.6%, and the number of execution cycles by 16.9%–59.4%.

Efficient synthesis of probabilistic quantum circuits with fallback

Recently it has been shown that Repeat-Until-Success (RUS) circuits can approximate a given single-qubit unitary with an expected number of $T$ gates of about $1/3$ of what is required by optimal, deterministic, ancilla-free decompositions over the Clifford+$T$ gate set. In this work, we introduce a more general and conceptually simpler circuit decomposition method that allows for synthesis into protocols that probabilistically implement quantum circuits over several universal gate sets including, but not restricted to, the Clifford+$T$ gate set. The protocol, which we call Probabilistic Quantum Circuits with Fallback (PQF), implements a walk on a discrete Markov chain in which the target unitary is an absorbing state and in which transitions are induced by multi-qubit unitaries followed by measurements. In contrast to RUS protocols, the presented PQF protocols terminate after a finite number of steps. Specifically, we apply our method to the Clifford+$T$, Clifford+$V$, and Clifford+$\pi/12$ gate sets to achieve decompositions with expected gate counts of $\log_b(1/\varepsilon)+O(\log(\log(1/\varepsilon)))$, where $b$ is a quantity related to the expansion property of the underlying universal gate set.

A Technology Based Complexity Model for Reversible Cuccaro Ripple-Carry Adder

Quantum computing and circuits are of growing interest and so is reversible logic as it plays an important role in the synthesis of quantum circuits. Moreover, reversible logic provides an alternative to classical computing machines, that may overcome many of the power dissipation problems in the near future. In effect, the applied adiabatic signals are known to allow the signal energy stored on the various capacitances of the circuit to be redistributed rather than being dissipated as heat. They additionally avoid calculation errors introduced by the use of conventional rectangular pulses. Some ripple-carry adders based on a do-spy-undo structure have been designed and tested reversibly. This paper presents a simple complexity model taking into account some physical aspects of the technology, from the study of a cascade of Cuccaro adders processed in standard 0.35 mC MOS technology and used in true reversible calculation (computations being performed forwards and backwards such that addition and subtraction are made reversibly with the same chip), through both, simulations and experimental results. This paper provides a simple physical complexity model as basis for future cost models.

Automatic synthesis of quantum circuits for point addition on ordinary binary elliptic curves

When designing quantum circuits for Shor's algorithm to solve the discrete logarithm problem, implementing the group arithmetic is a cost-critical task. We introduce a software tool for the automatic generation of addition circuits for ordinary binary elliptic curves, a prominent platform group for digital signatures. The resulting circuits reduce the number of $$T$$T-gates by a factor $$13/5$$13/5 compared to the best previous construction, without increasing the number of qubits or $$T$$T-depth. The software also optimizes the (CNOT) depth for $${\mathbb F}_2$$F2-linear operations by means of suitable graph colorings.

QLib

Quantum algorithms are known for their ability to solve some problems much faster than classical algorithms. They are executed on quantum circuits, which consist of a cascade of quantum gates. However, synthesis of quantum circuits is not straightforward because of the complexity of quantum algorithms. Generally, quantum algorithms contain two parts: classical and quantum. Thus, synthesizing circuits for the two parts separately reduces overall synthesis complexity. In addition, many quantum algorithms use similar subroutines that can be implemented with similar circuit modules. Because of their frequent use, it is important to use automated scripts to generate such modules efficiently. These modules can then be subjected to further synthesis optimizations. This article proposes QLib, a quantum module library, which contains scripts to generate quantum modules of different sizes and specifications for well-known quantum algorithms. Thus, QLib can also serve as a suite of benchmarks for quantum logic and physical synthesis.

Quantum circuit physical design methodology with emphasis on physical synthesis

In our previous works, we have introduced the concept of "physical synthesis" as a method to consider the mutual effects of quantum circuit synthesis and physical design. While physical synthesis can involve various techniques to improve the characteristics of the resulting quantum circuit, we have proposed two techniques (namely gate exchanging and auxiliary qubit selection) to demonstrate the effectiveness of the physical synthesis. However, the previous contributions focused mainly on the physical synthesis concept, and the techniques were proposed only as a proof of concept. In this paper, we propose a methodological framework for physical synthesis that involves all previously proposed techniques along with a newly introduced one (called auxiliary qubit insertion). We will show that the entire flow can be seen as one monolithic methodology. The proposed methodology is analyzed using a large set of benchmarks. Experimental results show that the proposed methodology decreases the average latency of quantum circuits by about 36.81 % for the attempted benchmarks.

Floating point representations in quantum circuit synthesis

We provide a non-deterministic quantum protocol that approximates the single qubit rotations Rx(2ϕ21ϕ22) using Rx(2ϕ1) and Rx(2ϕ2) and a constant number of Clifford and T operations. We then use this method to construct a ‘floating point’ implementation of a small rotation wherein we use the aforementioned method to construct the exponent part of the rotation and also to combine it with a mantissa. This causes the cost of the synthesis to depend more strongly on the relative (rather than absolute) precision required. We analyze the mean and variance of the T-count required to use our techniques and provide new lower bounds for the T-count for ancilla free synthesis of small single-qubit axial rotations. We further show that our techniques can use ancillas to beat these lower bounds with high probability. We also discuss the T-depth of our method and see that the vast majority of the cost of the resultant circuits can be shifted to parallel computation paths.

A Meet-in-the-Middle Algorithm for Fast Synthesis of Depth-Optimal Quantum Circuits

We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speed-up over simple brute force algorithms. As an illustration of our method we implemented this algorithm and found factorizations of the commonly used quantum logical operations into elementary gates in the Clifford+T set. In particular, we report a decomposition of the Toffoli gate over the set of Clifford and T gates. Our decomposition achieves a total T-depth of 3, thereby providing a 40% reduction over the previously best known decomposition for the Toffoli gate. Due to the size of the search space the algorithm is only practical for small parameters, such as the number of qubits, and the number of gates in an optimal implementation.

Synthesis of multivalued quantum logic circuits by elementary gates

We propose the generalized controlled X (GCX) gate as the two-qudit elementary gate, and based on Cartan decomposition, we also give the one-qudit elementary gates. Then we discuss the physical implementation of these elementary gates and show that it is feasible with current technology. With these elementary gates many important qudit quantum gates can be synthesized conveniently. We provide efficient methods for the synthesis of various kinds of controlled qudit gates and greatly simplify the synthesis of existing generic multi-valued quantum circuits. Moreover, we generalize the quantum Shannon decomposition (QSD), the most powerful technique for the synthesis of generic qubit circuits, to the qudit case. A comparison of ququart (d=4) circuits and qubit circuits reveals that using ququart circuits may have an advantage over the qubit circuits in the synthesis of quantum circuits.

A Review on Performances of Reversible Ripple-Carry Adders

Abstract Quantum computing and circuits are of growing interest and so is reversible logic as it plays an important role in the synthesis of quantum circuits. Moreover, reversible logic provides an alternative to classical computing machines, that may overcome many of the power dissipation problems in the near future. Some ripple-carry adders based on a do-spy-undo structure have been designed and tested reversibly. This paper presents a brief overview of the performances obtained with such chips processed in standard 0.35 μm CMOS technology and used in true reversible calculation (computations are performed forwards and backwards such that addition and subtraction are made reversibly with the same chip). Adiabatic signals used are known to allow the signal energy stored on the various capacitances of the circuit to be redistributed rather than being dissipated as heat while allowing to avoid calculation errors introduced by the use of conventional rectangular pulses. Through the example of both simulations and experimental results, this paper aims at providing a base of knowledge and knowhow in physical implementation of reversible circuits.

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.

Exact synthesis of three-qubit quantum circuits from non-binary quantum gates

Because of recent nano-technological advances, nano-structured systems have become highly ordered, making it quantum computing schemas possible. We propose an approach to optimally synthesise quantum circuits from non-permutative quantum gates such as controlled-square-root-of-not (i.e., controlled-V). Our approach reduces the synthesis problem to multiple-valued optimisation and uses group theory. We devise a novel technique that transforms the quantum logic synthesis problem from a multi-valued constrained optimisation problem to a permutable representation. The transformation enables us to use group theory to exploit the symmetric properties of the synthesis problem. Assuming a cost of one for each two-qubit gate, we found all reversible circuits with quantum costs of 4, 5, 6, etc., and give another algorithm to realise these reversible circuits with quantum gates. The approach can be used for both binary permutative deterministic circuits and probabilistic circuits such as controlled random-number generators and hidden Markov models.

REVERSIBLE VITERBI ALGORITHM AND ITS CLOSED-SYSTEM Q-DOMAIN CIRCUIT DESIGN AND COMPUTATION

Novel convolution-based multiple-stream error-control coding and decoding methods and their corresponding circuits are introduced. The new coding method applies the reversibility property in (1) the convolution-based encoder for multiple-stream error-control encoding and (2) in the new reversible Viterbi decoding algorithm for multiple-stream error-correction decoding. The complete synthesis of quantum circuits for the quantum realization of the new quantum Viterbi cell in the quantum domain (Q-domain) is also introduced, and the associated quantum computing representations and operations are presented. In quantum mechanics, a closed system is an isolated system that cannot exchange energy or matter with its surroundings and does not interact with other quantum systems. Closed quantum systems obey the unitary evolution and thus they are reversible. Reversibility property in error-control coding can be important for the following main reasons: (1) reversibility is a basic requirement for low-power circuit design in future technologies such as in closed-system quantum computing (QC), (2) reversibility leads to super-speedy encoding/decoding operations because of the superposition and entanglement properties that exist in the reversible closed-system quantum computing circuits and systems, and (3) the reversibility relationship between multiple-streams of data can be used for the correction of errors that are usually uncorrectable using the implemented decoding algorithm such as in the case of triple-errors that are uncorrectable using the irreversible Viterbi algorithm.

Synthesis of quantum circuits for $d$-level systems by using cosine-sine

We study the problem of designing minimal quantum circuits for any operations on $n$ qudits by means of the cosine-sine decomposition. Our method is based on a divide-and-conquer strategy. In that strategy, the size of the produced quantum circuit depends on whether the partitioning is balanced. We provide a new cosine-sine decomposition based on a balanced partitioning for $d$-level systems. The produced circuit is not asymptotically optimal except when $d$ is a power of two, but, when the number of qudits $n$ is small, our method can produce the smallest quantum circuit compared to the circuits produced by other synthesis methods. For example, when $d=3$ (three-level systems) and $n=2$ (two qudits), then the number of two-qudit operations called CINC, which is a generalized versions of CNOT, is 36 whereas the previous method needs 156 CINC gates. Moreover, we show that our method is useful for designing a polynomial-size quantum circuit for the radix-$d$ quantum Fourier transform.

Gate complexity using dynamic programming

The relationship between efficient quantum gate synthesis and control theory has been a topic of recent interest in the quantum computing literature. Motivated by this work, we describe how the dynamic programming technique from optimal control may be used in principle to determine gate complexity and for the optimal synthesis of quantum circuits. We illustrate the dynamic programming methodology using a simple example on the Lie group SU(2).