The Solovay-Kitaev algorithm

This introduction to quantum computing from a classical programmer's perspective is meant for students and practitioners alike. More than 50 quantum techniques and algorithms are explained with mathematical derivations and code for simulation, using an open-source code base in Python and C++. New material throughout this fully revised and expanded second edition includes new chapters on Quantum Machine Learning, State Preparation, and Similarity Tests. Coverage includes algorithms exploiting entanglement, black-box algorithms, the quantum Fourier transform, phase estimation, quantum walks, and foundational QML algorithms. Readers will find detailed, easy-to-follow derivations and implementations of Shor's algorithm, Grover's algorithm, SAT3, graph coloring, the Solovay-Kitaev algorithm, Moettoenen's algorithm, quantum mean, median, and minimum finding, Deutsch's algorithm, Bernstein-Vazirani, quantum teleportation and superdense coding, the CHSH game, and, from QML, the HHL algorithm, Euclidean distance, and PCA. The book also discusses productivity issues like quantum noise, error correction, quantum programming languages, compilers, and techniques for transpilation.

Quantum Gates and Quantum Circuits of Stock Portfolio

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 algorithm to the case of topological compiling of Fibonacci anyons and obtain near-optimal braiding sequences for arbitrary single-qubit unitaries. Our algorithm may carry over to other challenging quantum discrete problems, thus opening up a new avenue for intriguing applications of deep learning in quantum physics.

The classical computer is restricted by Moore's law and other factors into the development bottleneck, quantum computer by virtue of its natural acceleration advantage into the field of vision. Quantum computing can achieve the acceleration effect compared with classical computing in many fields. In the field of integer decomposition, Shor algorithm can be solved in polynomial time, which breaks the security of RSA public key cryptosystem. However, the quantum bits (logn + 2, where n is the integer to be decomposed) required by Shor algorithm are difficult to be realized on today's quantum computers. Therefore, the hybrid quantum algorithm model of "classical + quantum (Grover algorithm)" is tried to realize the decomposition of large integers. In this paper, based on the previous hybrid quantum algorithm model, considering that the continuous accumulation of errors in the circuit of the quantum algorithm will lead to the failure of the algorithm as the depth increases, part of the circuit of the quantum algorithm oracle is optimized to reduce the consumption of the depth, quantum gate and quantum bit in the circuit of the quantum algorithm, so as to make it easier to run on the actual quantum computer. By running the optimized quantum circuit on the back end of the simulator of IBM Q quantum cloud platform, and trying to add thermal relaxation and depolarization noises to the circuit, the influence of different noises on the circuit under different iterations was explored.

We provide a pedagogical presentation of Shor's factoring algorithm, which is a quantum algorithm for factoring very large numbers (of order of hundreds to thousands of bits) in polynomial time. In contrast, all known classical algorithms for the factoring problem take an exponential time to factor such large numbers. Shor's algorithm therefore has profound implication for public-key encryption such as RSA and Diffie–Hellman key exchange. We assume no prior knowledge of Shor's algorithm beyond a basic familiarity with the circuit model of quantum computing. Shor's algorithm contains a number of moving parts, and can be rather daunting at first. The literature is replete with derivations and expositions of Shor's algorithm, but most of them seem to be lacking in essential details, and none of them provide a pedagogical presentation. They require a thicket of appendices and assume a knowledge of quantum algorithms and classical mathematics with which the reader might not be familiar. We therefore start with first principle derivations of the quantum Fourier transform (QFT) and quantum phase estimation (QPE), which are the essential building blocks of Shor's algorithm. We then go on to develop the theory of modular exponentiation (ME) operators, one of the fundamental components of Shor's algorithm, and the place where most of the quantum resources are deployed. We also delve into the number theory that establishes the link between factorization and the period of the modular exponential function. We then apply the QPE algorithm to obtain Shor's factoring algorithm. We also discuss the post-quantum processing and the method of continued fractions, which is used to extract the exact period of the modular exponential function from the approximately measured phase angles of the ME operator. The manuscript then moves on to a series of examples. We first verify the formalism by factoring N=15, the smallest number accessible to Shor's algorithm. We then proceed to factor larger integers, developing a systematic procedure that will find the ME operators for any semi-prime N=p×q (where q and p are prime). Finally, we factor the composite numbers N=21, 33, 35, 143, 247 using the Qiskit simulator. It is observed that the ME operators are somewhat forgiving, and truncated approximate forms are able to extract factors just as well as the exact operators. This is because the method of continued fractions only requires an approximate phase value for its input, which suggests that implementing Shor's algorithm might not be as difficult as first suspected. Quanta 2023; 12: 41–130.

Non-adiabatic holonomic quantum gate in decoherence-free subspaces is of greatly practical importance due to its built-in fault tolerance, coherence stabilization virtues, and short run-time. Here, we propose some compact schemes to implement two- and three-qubit controlled unitary quantum gates and Fredkin gate. For the controlled unitary quantum gates, the unitary operator acting on the target qubit is an arbitrary single-qubit gate operation. The controlled quantum gates can be directly implemented by utilizing non-adiabatic holonomy in decoherence-free subspaces and the required resource for the decoherence-free subspace encoding is minimal by using only two neighboring physical qubits undergoing collective dephasing to encode a logical qubit.

IBM’s quantum computers implement gates from Clifford +T gate library. All single qubit gates are implemented, but only a subset of the possible CNOT are provided. It is well known that the functionally of the missing gates can be achieved by a sequence of gates. The sequence of gates is based on SWAP gates. Up to seven elementary gates are required to implement a SWAP gate. In this paper we show how the same effect can be achieved with fewer gates. To show the potential of the proposed transformations, an example is presented where a reduction of 44% in the gate count and a 26% reduction in the number of levels for IBM’s QX5 computer is achieved. An algorithm that is considered state of the art, is used for the comparison.

Quantum Fourier Transform (QFT) is a key part of quantum algorithms such as Shor algorithm, Grover algorithm, and quantum phase estimation algorithm. The “Songshan” supercomputer system adopts the heterogeneous architecture of domestic CPU+DCU. The simulation implementation of the quantum Fourier transform on the “Songshan” supercomputer can promote the research and optimization of related quantum algorithms. Based on the HIP heterogeneous programming model, this paper first maps the phase shift transformation R gate operation of the QFT transformation to the DCU accelerator, and initially constructs the heterogeneous implementation of QFT on the “Songshan” supercomputer; then, the Hadamard gate transformation operation originally implemented on the host side was implemented on the DCU, which greatly reduces the communication overhead caused by the quantum information data transmission between the CPU and the DCU device; finally, the quantum logic gate is analyzed, and based on the operation of the phase shift transformation R gate on the probability amplitude information in the simulation algorithm, an optimization scheme to improve the activity of the DCU thread is proposed. During the experimental verification process, the realized 28-qubit quantum Fourier transform optimization has achieved 13.440 speedup compared with the serially realized QFT simulation on the CPU. The basic quantum gate optimization method based on DCU acceleration equipment in this paper can provide a reference for the efficient simulation of other quantum algorithms on domestic heterogeneous platforms.

Approximation errors must be taken into account when compiling quantum programs into a low-level gate set. We present a methodology that tracks such errors automatically and then optimizes accuracy parameters to guarantee a specified overall accuracy while aiming to minimize the implementation cost in terms of quantum gates. The core idea of our approach is to extract functions that specify the optimization problem directly from the high-level description of the quantum program. Then, custom compiler passes optimize these functions, turning them into (near-)symbolic expressions for (1) the total error and (2) the implementation cost (e.g., total quantum gate count). All unspecified parameters of the quantum program will show up as variables in these expressions, including accuracy parameters. After solving the corresponding optimization problem, a circuit can be instantiated from the found solution. We develop two prototype implementations, one in C++ based on Clang/LLVM, and another using the Q# compiler infrastructure. We benchmark our prototypes on typical quantum computing programs, including the quantum Fourier transform, quantum phase estimation, and Shor's algorithm.

The publication in 1994 of Shor's algorithm, which allows factorization of composite number N in a time polynomial in its binary length L has been the primary catalyst for the race to construct a functional quantum computer. However, it seems clear that any practical system that may be developed will not be able to perform completely error free quantum gate operations or shield even idle qubits from inevitable error effects. Hence, the practicality of quantum algorithms needs to be investigated to estimate what demands must be made of quantum error correction (QEC). Several different quantum circuits implementing the quantum period finding (QPF) subroutine, which lies at the heart of Shor's algorithm have been designed, but each tacitly assumes that arbitrary pairs of qubits can be interacted. While some architectures posses this property, many promising proposals are best suited to realizing a single line of qubits with nearest neighbor interactions only. This paper will present a circuit suitable for implementing the QPF subroutine for such linear nearest neighbor (LNN) designs. We will then present direct simulation results showing for both the LNN circuit and for a circuit utilizing arbitrary interactions, that the QPF subroutine is very sensitive to a small number of errors in the entire circuit. These results can then be used to briefly examine some of the practical issues to implementing such large scale quantum algorithms.

Quantum computing algorithms have been shown to produce performant quantum kernels for machine-learning classification problems. Here, we examine the performance of quantum kernels for regression problems of practical interest. For an unbiased benchmarking of quantum kernels, it is necessary to construct the most optimal functional form of the classical kernels and the most optimal quantum kernels for each given data set. We develop an algorithm that uses an analog of the Bayesian information criterion to optimize the sequence of quantum gates used to estimate quantum kernels for Gaussian process models. The algorithm increases the complexity of the quantum circuits incrementally, while improving the performance of the resulting kernels, and is shown to yield much higher model accuracy with fewer quantum gates than a fixed quantum circuit ansatz. We demonstrate that quantum kernels thus obtained can be used to build accurate models of global potential energy surfaces (PES) for polyatomic molecules. The average interpolation error of the six-dimensional PES obtained with a random distribution of 2000 energy points is 16 cm−1 for H3O+, 15 cm−1 for H2CO and 88 cm−1 for HNO2. We show that a compositional optimization of classical kernels for Gaussian process regression converges to the same errors. This indicates that quantum kernels can achieve the same, though not better, expressivity as classical kernels for regression problems.

Ramsey's theorem [5] asserts that every infinite set X has the following partition property (RP): For every partition of the set [X]2 of two-element subsets of X into two pieces, there is an infinite subset Y of X such that [Y]2 is included in one of the pieces. Ramsey explicitly indicated that his proof of this theorem used the axiom of choice. Kleinberg [3] showed that every proof of Ramsey's theorem must use the axiom of choice, although rather weak forms of this axiom suffice. J. Dawson has raised the question of the position of Ramsey's theorem in the hierarchy of weak axioms of choice.In this paper, we prove or refute the provability of each of the possible implications between Ramsey's theorem and the weak axioms of choice mentioned in Appendix A.3 of Jech's book [2]. Our results, along with some known facts which we include for completeness, may be summarized as follows (the notation being as in [2]):A. The following principles do not (even jointly) imply Ramsey's theorem, nor does Ramsey's theorem imply any of them:the Boolean prime ideal theorem,the selection principle,the order extension principle,the ordering principle,choice from wellordered sets (ACW),choice from finite sets,choice from pairs (C2).B. Each of the following principles implies Ramsey's theorem, but none of them follows from Ramsey's theorem:the axiom of choice,wellordered choice (∀kACk),dependent choice of any infinite length k (DCk),countable choice (ACN0),nonexistence of infinite Dedekind-finite sets (WN0).

Improving gate performance is vital for scalable quantum computing. Universal quantum computing also requires gate fidelity to reach a high level. For a superconducting quantum processor, which operates in the microwave band, the single-qubit gates are usually realized with microwave driving. The crosstalk between microwave pulses is a non-negligible error source. In this article, we propose an error mitigation scheme to address this crosstalk issue for single-qubit gates. There are three steps in our method. First, by controlling the detuning between qubits, the microwave induced classical crosstalk error can be constrained within the computational subspace. Second, by applying the general decomposition procedure, the arbitrary single-qubit gate can be decomposed as a sequence of X and virtual Z gates. Finally, by optimizing the parameters in virtual Z gates, the error constrained in the computational space can be corrected. Using our method, no additional compensation signals are needed, arbitrary single-qubit gate time will not be prolonged, and the circuit depth containing simultaneous single-qubit gates will also not increase. The simulation results show that, in a specific regime of qubit–qubit detuning, the infidelities of simultaneous single-qubit gates can be as low as that without microwave crosstalk.

The security of public-key cryptography relies on well-studied hard problems, problems for which we do not have efficient algorithms. Factorization and discrete logarithm are the two most known and used hard problems. Unfortunately, they can be easily solved on a quantum computer by Shor's algorithm. Also, the research area of cryptography demands for crypto-diversity which says that we should offer a range of hard problems for public-key cryptography. If one hard problem proves to be easy, we should be able to provide alternative solutions. Some of the candidates for post-quantum hard problems, i.e. problems which are believed to be hard even on a quantum computer, are the Learning Parity with Noise (LPN), the Learning with Errors (LWE) and the Shortest Vector Problem (SVP). A thorough study of these problems is needed in order to assess their hardness. In this thesis we focus on the algorithmic study of LPN. LPN is a hard problem that is attractive, as it is believed to be post-quantum resistant and suitable for lightweight devices. In practice, it has been employed in several encryption schemes and authentication protocols. At the beginning of this thesis, we take a look at the existing LPN solving algorithms. We provide the theoretical analysis that assesses their complexity. We compare the theoretical results with practice by implementing these algorithms. We study the efficiency of all LPN solving algorithms which allow us to provide secure parameters that can be used in practice. We push further the state of the art by improving the existing algorithms with the help of two new frameworks. In the first framework, we split an LPN solving algorithm into atomic steps. We study their complexity, how they impact the other steps and we construct an algorithm that optimises their use. Given an LPN instance that is characterized by the noise level and the secret size, our algorithm provides the steps to follow in order to solve the instance with optimal complexity. In this way, we can assess if an LPN instance provides the security we require and we show what are the secure instances for the applications that rely on LPN. The second framework handles problems that can be decomposed into steps of equal complexity. Here, we assume that we have an adversary that has access to a finite or infinite number of instances of the same problem. The goal of the adversary is to succeed in just one instance as soon as possible. Our framework provides the strategy that achieves this. We characterize an LPN solving algorithm in this framework and show that we can improve its complexity in the scenario where the adversary is restricted. We show that other problems, like password guessing, can be modeled in the same framework.

The standard approach to fault-tolerant quantum computation is to store\ninformation in a quantum error correction code, such as the surface code, and\nprocess information using a strategy that can be summarized as\ndistill-then-synthesize. In the distill step, one performs several rounds of\ndistillation to create high-fidelity logical qubits in a magic state. Each such\nmagic state provides one good T gate. In the synthesize step, one seeks the\noptimal decomposition of an algorithm into a sequence of many T gates\ninterleaved with Clifford gates. This gate-synthesis problem is well understood\nfor multiqubit gates that do not use any Hadamards. We present an in-depth\nanalysis of a unified framework that realises one round of distillation and\nmultiqubit gate synthesis in a single step. We call these synthillation\nprotocols, and show they lead to a large reduction in resource overheads. This\nis because synthillation can implement a general class of circuits using the\nsame number of T-states as gate synthesis, yet with the benefit of quadratic\nerror suppression. This general class includes all circuits primarily dominated\nby control-control-Z gates, such as adders and modular exponentiation routines\nused in Shor's algorithm. Therefore, synthillation removes the need for a\ncostly round of magic state distillation. We also present several additional\nresults on the multiqubit gate-synthesis problem. We provide an efficient\nalgorithm for synthesizing unitaries with the same worst-case resource scaling\nas optimal solutions. For the special case of synthesizing\ncontrolled-unitaries, our techniques are not just efficient but exactly\noptimal. We observe that the gate-synthesis cost, measured by T-count, is often\nstrictly subadditive. Numerous explicit applications of our techniques are also\npresented.\n