Here we describe a simple mechanical procedure for compiling a quantum gate network into the natural gates (pulses and delays) for an Ising quantum computer. The aim is not necessarily to generate the most efficient pulse sequence, but rather to develop an efficient compilation algorithm that can be easily implemented in large spin systems. The key observation is that it is not always necessary to refocus all the undesired couplings in a spin system. Instead the coupling evolution can simply be tracked and then corrected at some later time. Although described within the language of NMR the algorithm is applicable to any design of quantum computer based on Ising couplings.
Quantum algorithms require a universal set of gates that can be implemented in a physical system. For these, an optimal decomposition into a sequence of available operations is desired. Here, we present a method to find such sequences for a small-scale ion trap quantum information processor. We further adapt the method to state preparation and quantum algorithms with in-sequence measurements.
Given a set of quantum gates and a target unitary operation, the most elementary task of quantum compiling is the identification of a sequence of the gates that approximates the target unitary to a determined precision $\varepsilon$. Solovay-Kitaev theorem provides an elegant solution which is based on the construction of successively tighter `nets' around the unity comprised by successively longer sequences of gates. The procedure for the construction of the nets, according to this theorem, requires accessibility to the inverse of the gates as well. In this work, we propose a method for const…
The design and optimization of quantum circuits is central to quantum computation. This paper presents new algorithms for compiling arbitrary 2^n x 2^n unitary matrices into efficient circuits of (n-1)-controlled single-qubit and (n-1)-controlled-NOT gates. We first present a general algebraic optimization technique, which we call the Palindrome Transform, that can be used to minimize the number of self-inverting gates in quantum circuits consisting of concatenations of palindromic subcircuits. For a fixed column ordering of two-level decomposition, we then give an numerative algorithm for min…
The architecture of circuital quantum computers requires computing layers devoted to compiling high-level quantum algorithms into lower-level circuits of quantum gates. The general problem of quantum compiling is to approximate any unitary transformation that describes the quantum computation, as a sequence of elements selected from a finite base of universal quantum gates. The existence of an approximating sequence of one qubit quantum gates is guaranteed by the Solovay-Kitaev theorem, which implies sub-optimal algorithms to establish it explicitly. Since a unitary transformation may require …
Variational hybrid quantum-classical algorithms (VHQCAs) are near-term algorithms that leverage classical optimization to minimize a cost function, which is efficiently evaluated on a quantum computer. Recently VHQCAs have been proposed for quantum compiling, where a target unitary $U$ is compiled into a short-depth gate sequence $V$. In this work, we report on a surprising form of noise resilience for these algorithms. Namely, we find one often learns the correct gate sequence $V$ (i.e., the correct variational parameters) despite various sources of incoherent noise acting during the cost-eva…
Randomized compiling reduces the effects of errors on quantum computers by tailoring arbitrary Markovian errors into stochastic Pauli noise. Here we prove that randomized compiling also tailors non-Markovian errors into local stochastic Pauli noise and investigate the technique's limitations. We show through analysis and numerical results that randomized compiling alters errors in three distinct helpful ways. First, it prevents the coherent accumulation of errors (including hard to remove crosstalk effects) across gate cycles by destroying intercycle coherent correlations. Second, it converts …
Compiling quantum circuits lends itself to an elegant formulation in the language of rewriting systems on non commutative polynomial algebras $\mathbb Q\langle X\rangle$. The alphabet $X$ is the set of the allowed hardware 2-qubit gates. The set of gates that we wish to implement from $X$ are elements of a free monoid $X^*$ (obtained by concatenating the letters of $X$). In this setting, compiling an idealized gate is equivalent to computing its unique normal form with respect to the rewriting system $\mathcal R\subset \mathbb Q\langle X\rangle$ that encodes the hardware constraints and capabi…
We propose a method of compiling that permits to identify quantum circuits able to simulate arbitrary $n$-qubit unitary operations via the adjustment of angles in single-qubit gates therein. The method of compiling itself extends older quantum control techniques and stays computationally tractable for several qubits. As an application we identify compiling universal circuits for $3$, $4$ and $5$ qubits consisting of $16$, $64$ and $ 256$ CNOTs respectively.
Quantum compiling addresses the problem of approximating an arbitrary quantum gate with a string of gates drawn from a particular finite set. It has been shown that this is possible for almost all choices of base sets and furthermore that the number of gates required for precision epsilon is only polynomial in log 1/epsilon. Here we prove that using certain sets of base gates quantum compiling requires a string length that is linear in log 1/epsilon, a result which matches the lower bound from counting volume up to constant factor.
Topological quantum computation is an implementation of a quantum computer in a way that radically reduces decoherence. Topological qubits are encoded in the topological evolution of two-dimensional quasi-particles called anyons and universal set of quantum gates can be constructed by braiding these anyons yielding to a topologically protected circuit model. In the present study we remind the basics of this emerging quantum computation scheme and illustrate how a topological qubit built with three Fibonacci anyons might be adopted to achieve leakage free braiding gate by exchanging the anyons …
We present an algorithm for compiling arbitrary unitaries into a sequence of gates native to a quantum processor. As accurate CNOT gates are hard for the foreseeable Noisy- Intermediate-Scale Quantum devices era, our A* inspired algorithm attempts to minimize their count, while accounting for connectivity. We discuss the search strategy together with metrics to expand the solution frontier. For a workload of circuits with complexity appropriate for the NISQ era, we produce solutions well within the best upper bounds published in literature and match or exceed hand tuned implementations, as wel…
Compiling quantum algorithms for near-term quantum computers (accounting for connectivity and native gate alphabets) is a major challenge that has received significant attention both by industry and academia. Avoiding the exponential overhead of classical simulation of quantum dynamics will allow compilation of larger algorithms, and a strategy for this is to evaluate an algorithm's cost on a quantum computer. To this end, we propose a variational hybrid quantum-classical algorithm called quantum-assisted quantum compiling (QAQC). In QAQC, we use the overlap between a target unitary $U$ and a …
College of Mathematics and Informatics, South China Agricultural University（华南农业大学数学与信息学院）
Quantum compiling aims to construct a quantum circuit V by quantum gates drawn from a native gate alphabet, which is functionally equivalent to the target unitary U. It is a crucial stage for the running of quantum algorithms on noisy intermediate-scale quantum (NISQ) devices. However, the space for structure exploration of quantum circuit is enormous, resulting in the requirement of human expertise, hundreds of experimentations or modifications from existing quantum circuits. In this paper, we propose a variational quantum compiling (VQC) algorithm based on reinforcement learning (RL), in ord…
Efficiently processing basic linear algebra subroutines is of great importance for a wide range of computational problems. In this paper, we consider techniques to implement matrix functions on a quantum computer, which are composed of basic matrix operations on a set of matrices. These matrix operations include addition, multiplication, Kronecker sum, tensor product, Hadamard product, and single-matrix functions. We discuss the composed matrix functions in terms of the estimation of scalar quantities such as inner products, trace, determinant and Schatten p-norms. We thus provide a framework …
Quantum computers are poised to radically outperform their classical counterparts by manipulating coherent quantum systems. A realistic quantum computer will experience errors due to the environment and imperfect control. When these errors are even partially coherent, they present a major obstacle to achieving robust computation. Here, we propose a method for introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged. We prove that this randomization tailors the noise into stochastic Pauli errors, leading to dra…
In the context of variational compiling of a continuous-variable (CV) unitary operation, the architecture and parameter space of the Xanadu X8 processor constrain both the set of feasible compiling problems and the allowed cost functions. In this paper, we motivate a faithful cost function for variational compiling of a two-mode, continuous-variable beamsplitter gate with real matrix elements (i.e., a "zero-phase" beamsplitter) that complies with the constraints of the X8 processor. This cost function is then computed on the X8. Despite the noise in the processor, we find that the cost functio…
Tsinghua University（清华大学）；University of California, San Diego（加利福尼亚大学圣地亚哥分校）；Peking University（北京大学）；Shanghai Qi Zhi Institute（上海期智研究院）
Quantum compiling, a process that decomposes the quantum algorithm into a series of hardware-compatible commands or elementary gates, is of fundamental importance for quantum computing. We introduce an efficient algorithm based on deep reinforcement learning that compiles an arbitrary single-qubit gate into a sequence of elementary gates from a finite universal set. It generates near-optimal gate sequences with given accuracy and is generally applicable to various scenarios, independent of the hardware-feasible universal set and free from using ancillary qubits. For concreteness, we apply this…
We develop the first constructive algorithms for compiling single-qubit unitary gates into circuits over the universal $V$ basis. The $V$ basis is an alternative universal basis to the more commonly studied $\{H,T\}$ basis. We propose two classical algorithms for quantum circuit compilation: the first algorithm has expected polynomial time (in precision $\log(1/ε)$) and offers a depth/precision guarantee that improves upon state-of-the-art methods for compiling into the $\{H,T\}$ basis by factors ranging from 1.86 to $\log_2(5)$. The second algorithm is analogous to direct search and yields ci…
We study the problem of finding the best approximate circuit that is the closest (in some pertinent metric) to a target circuit, and which satisfies a number of hardware constraints, like gate alphabet and connectivity. We look at the problem in the CNOT+rotation gate set from a mathematical programming standpoint, offering contributions both in terms of understanding the mathematics of the problem and its efficient solution. Among the results that we present, we are able to derive a 14-CNOT 4-qubit Toffoli decomposition from scratch, and show that the Quantum Shannon Decomposition can be comp…
Current quantum computers are especially error prone and require high levels of optimization to reduce operation counts and maximize the probability the compiled program will succeed. These computers only support operations decomposed into one- and two-qubit gates and only two-qubit gates between physically connected pairs of qubits. Typical compilers first decompose operations, then route data to connected qubits. We propose a new compiler structure, Orchestrated Trios, that first decomposes to the three-qubit Toffoli, routes the inputs of the higher-level Toffoli operations to groups of near…
We propose a general and systematic strategy to compile arbitrary quantum channels without using ancillary qubits, based on proximal policy optimization -- a powerful deep reinforcement learning algorithm. We rigorously prove that, in sharp contrast to the case of compiling unitary gates, it is impossible to compile an arbitrary channel to arbitrary precision with any given finite elementary channel set, regardless of the length of the decomposition sequence. However, for a fixed accuracy $ε$ one can construct a universal set with constant number of $ε$-dependent elementary channels, such that…
A method for compiling quantum algorithms into specific braiding patterns for non-Abelian quasiparticles described by the so-called Fibonacci anyon model is developed. The method is based on the observation that a universal set of quantum gates acting on qubits encoded using triplets of these quasiparticles can be built entirely out of three-stranded braids (three-braids). These three-braids can then be efficiently compiled and improved to any required accuracy using the Solovay-Kitaev algorithm.
The success of the current generation of Noisy Intermediate-Scale Quantum (NISQ) hardware shows that quantum hardware may be able to tackle complex problems even without error correction. One outstanding issue is that of coherent errors arising from the increased complexity of these devices. These errors can accumulate through a circuit, making their impact on algorithms hard to predict and mitigate. Iterative algorithms like Quantum Imaginary Time Evolution are susceptible to these errors. This article presents the combination of both noise tailoring using Randomized Compiling and error mitig…
I describe the use of techniques based on composite rotations to develop controlled phase gates in which the effects of weak Ising couplings are suppressed. A tailored composite phase gate is described which both suppresses weak couplings and is relatively insensitive to systematic errors in the size of strong couplings.
We present a constructive method to create quantum circuits that implement oracles $|x\rangle|y\rangle|0\rangle^k \mapsto |x\rangle|y \oplus f(x)\rangle|0\rangle^k$ for $n$-variable Boolean functions $f$ with low $T$-count. In our method $f$ is given as a 2-regular Boolean logic network over the gate basis $\{\land, \oplus, 1\}$. Our construction leads to circuits with a $T$-count that is at most four times the number of AND nodes in the network. In addition, we propose a SAT-based method that allows us to trade qubits for $T$ gates, and explore the space/complexity trade-off of quantum circui…
Quantum compiling fills the gap between the computing layer of high-level quantum algorithms and the layer of physical qubits with their specific properties and constraints. Quantum compiling is a hybrid between the general-purpose compilers of computers, transforming high-level language to assembly language and hardware synthesis by hardware description language, where functions are automatically synthesized into customized hardware. Here we review the quantum compiling stack of both gate model quantum computers and the adiabatic quantum computers, respectively. The former involves low level …
I describe the use of techniques based on composite rotations to combat systematic errors in controlled phase gates, which form the basis of two qubit quantum logic gates. Although developed and described within the context of Nuclear Magnetic Resonanace (NMR) quantum computing these sequences should be applicable to any implementation of quantum computation based on Ising couplings. In combination with existing single qubit gates this provides a universal set of robust quantum logic gates.
The physical implementation of the quantum Control-Not gate for a two-spin system is investigated numerically. The concept of a generalized quantum Control-Not gate, with arbitrary phase shift, is introduced. It is shown that a resonant $π$-pulse provides a simple example of a generalized quantum Control-Not gate.
We exactly construct one- and two-qubit holonomic quantum gates in terms of isospectral deformations of an Ising model Hamiltonian. A single logical qubit is constructed out of two spin-1/2 particles; the qubit is a dimer. We find that the holonomic gates obtained are discrete but dense in the unitary group. Therefore an approximate gate for a desired one can be constructed with arbitrary accuracy.
We construct a minimal robust controlled-NOT gate with an Ising-type interaction by which elementary two-qubit gates are implemented. It is robust against inaccuracy of the coupling strength and the obtained quantum circuits are constructed with the minimal number (N=3) of elementary two-qubit gates and several one-qubit gates. It is noteworthy that all the robust circuits can be mapped to one-qubit circuits robust against a pulse length error. We also prove that a minimal robust SWAP gate cannot be constructed with N=3, but requires N=6 elementary two-qubit gates.
The gate version of quantum computation exploits several quantum key resources as superposition and entanglement to reach an outstanding performance. In the way, this theory was constructed adopting certain supposed processes imitating classical computer gates. As for optical as well as magnetic systems, those gates are obtained as quantum evolutions. Despite, in certain cases they are attained as an asymptotic series of evolution effects. The current work exploits the direct sum of the evolution operator on a non-local basis for the driven bipartite Heisenberg-Ising model to construct a set o…
We present protocols for implementation of universal quantum gates on an arbitrary superposition of quantum states in a scalable solid-state Ising spin quantum computer. The spin chain is composed of identical spins 1/2 with the Ising interaction between the neighboring spins. The selective excitations of the spins are provided by the gradient of the external magnetic field. The protocols are built of rectangular radio-frequency pulses. The phase and probability errors caused by unwanted transitions are minimized and computed numerically.
We consider the model of quantum computer, which is represented as a Ising spin lattice, where qubits (spin-half systems) are separated by the isolators (two spin-half systems). In the idle mode or at the single bit operations the total spin of isolators is 0. There are no need of complicated protocols for correcting the phase and probability errors due to permanent interaction between the qubits. We present protocols for implementation of universal quantum gates with the rectangular radio-frequency pulses.
The successful implementation of algorithms on quantum processors relies on the accurate control of quantum bits (qubits) to perform logic gate operations. In this era of noisy intermediate-scale quantum (NISQ) computing, systematic miscalibrations, drift, and crosstalk in the control of qubits can lead to a coherent form of error which has no classical analog. Coherent errors severely limit the performance of quantum algorithms in an unpredictable manner, and mitigating their impact is necessary for realizing reliable quantum computations. Moreover, the average error rates measured by randomi…
Quantum computation is a continuously growing research area which is based on nature and resources of quantum mechanics, as superposition and entanglement. In its quantum circuits version, the use of convenient and appropriate gates is essential. But while those gates adopt convenient forms for computational algorithms, their design depends on specific quantum systems and stu? being used. These gates need manage quantum systems based on physical interactions ruled by quantum Hamiltonians. With this, gates design is restricted to properties and limitations of interactions and physical elements …
Compiling quantum circuits to realistic hardware architectures using temporal planners
NASA Ames Research Center, Quantum Artificial Intelligence Laboratory（美国国家航空航天局阿姆斯研究中心，量子人工智能实验室）；USRA Research Institute for Advanced Computer Science (RIACS)（美国宇航局高级计算机科学研究所（RIACS））；Stinger Ghaffarian Technologies (SGT Inc.)（Stinger Ghaffarian技术（SGT公司））；NASA Ames Research Center, Planning and Scheduling Group（美国国家航空航天局阿姆斯研究中心，计划与调度组）
It is imperative to compile quantum circuits for Noisy Intermediate-Scale Quantum (NISQ) devices because of the limited connectivity of physical qubits and the high error rates of gate operations. One of the most critical steps in quantum circuit compilation is qubit routing, an NP-Hard problem that involves placing and moving logical qubits to minimize compilation overhead. In this study, we propose ISing mAchine Assisted Quantum compiler (ISAAQ) to perform qubit routing with Ising machines, which can efficiently solve Quadratic Unconstrained Binary Optimization (QUBO) problems. ISAAQ accurat…
A quantum computer consists of a set of quantum bits upon which operations called gates are applied to perform computations. In order to perform quantum algorithms, physicists would like to design arbitrary gates to apply to quantum bits. However, the physical limitations of the quantum computing device restrict the set of gates that physicists are able to apply. Thus, they must compose a sequence of gates from the permitted gate set, which approximates the gate they wish to apply - a process called quantum compiling. Austin Fowler proposes a method that finds optimal gate sequences in exponen…
Quantum circuit compilation with constant overhead
Google Quantum AI（谷歌量子人工智能）；Stanford University（斯坦福大学）；MIT（麻省理工学院）；Institute for Quantum Computing, University of Waterloo and Perimeter Institute for Theoretical Physics（滑铁卢大学量子计算研究所与佩里尔理论物理研究所）
This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of $π/ 2^k$, into an efficient fault-tolerant form using only Hadamard, controlled-{\sc not}, and $π/ 8$ gates. The algorithm runs in $O(\log^{2.71}(1/ε))$ time, and produces as output a sequence of $O(\log^{3.97}(1/ε))$ quantum gates which is guaranteed to approximate the desired quantum …
Current quantum processors are noisy, have limited coherence and imperfect gate implementations. On such hardware, only algorithms that are shorter than the overall coherence time can be implemented and executed successfully. A good quantum compiler must translate an input program into the most efficient equivalent of itself, getting the most out of the available hardware. In this work, we present novel deterministic algorithms for compiling recurrent quantum circuit patterns in polynomial time. In particular, such patterns appear in quantum circuits that are used to compute the ground state p…
As we are approaching actual application of quantum technology, it is essential to exploit the current quantum resources in the best possible way. With this in mind, it might not be beneficial to use the usual standard gate sets, inspired from classical logic gates, when compiling quantum algorithms when other less standardized gates currently perform better. We, therefore, consider a promising native gate, which occurs naturally in superconducting circuits, known as the diamond gate. We show how the diamond gate can be decomposed into standard gates and, using single-qubit gates, can work as …
The quantum circuit model allows gates between any pair of qubits yet physical instantiations allow only limited interactions. We address this problem by providing an interaction graph together with an efficient method for compiling quantum circuits so that gates are applied only locally. The graph requires each qubit to interact with 4 other qubits and yet the time-overhead for implementing any n-qubit quantum circuit is 6 log n. Building a network of quantum computing nodes according to this graph enables the network to emulate a single monolithic device with minimal overhead.
