Enabling accuracy-aware Quantum compilers using symbolic resource estimation

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.

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.

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.

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.

We analyze simple quantum error detection and quantum error correction protocols relevant to current experiments with superconducting qubits. We show that for qubits with energy relaxation the repetitive $N$-qubit codes cannot be used for quantum error correction, but can be used for quantum error detection. In the latter case it is sufficient to use only two qubits for the encoding. In the analysis we demonstrate a useful technique of unraveling the qubit energy relaxation into ``relaxation'' and ``no relaxation'' scenarios. Also, we propose and numerically analyze several two-qubit algorithms for quantum error detection and correction, which can be readily realized at the present-day level of the phase qubit technology.

Quantum Gates and Quantum Circuits of Stock Portfolio

In this study, the authors categorise all of the feasible constructions for the composite Galois field GF(((22)2)2) Advanced Encryption Standard (AES) S-box into four main architectures by their field representations and their algebraic properties. For each of the categories, a new optimisation scheme which exploits algebraic normal form representation followed by a sub-structure sharing optimisation is presented. This is performed by converting the subfield GF((22) inversion into several logical expressions, which will be in turn reduced using a common sub-expression elimination algorithm. The authors show that this technique can effectively reduce the total area gate count as well as the critical path gate count in composite field AES S-boxes. The resulting architecture that achieves maximum reduction in both total area coverage and critical path gate count is found and reported. The hardware implementations of the authors proposed AES S-boxes, along with their performance and cost are presented and discussed.

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.

This chapter reviews the general problems and the high-level operation of the major quantum algorithms. It describes some of the quantum algorithms in more detail. These algorithms include: Deutsch algorithm, Deutsch–Jozsa algorithm, Simon's algorithm, Shor's algorithm, quantum phase estimation algorithm, Grover's Quantum Search Algorithm (QSA), Boyer–Brassard–Høyer–Tapp QSA, Dürr–Høyer QSA, quantum counting algorithm, quantum heuristic algorithm, quantum genetic algorithm, Harrow–Hassidim–Lloyd algorithm, quantum mean algorithm, and quantum weighted sum algorithm. The chapter provides more physical insights into some of the algorithms. Quantum computation is based on quantum interference, which is a dynamical process that allows one to evolve initial quantum states (inputs) into final states (outputs) by modifying intermediate multi-particle superpositions in some prescribed way. The chapter illustrates how interference patterns lead to computational problems that are well suited to quantum computations, by presenting the first such problem that was proposed by David Deutsch.

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

This project builds toward a comprehensive error-mitigating toolkit that makes quantum programming more robust and adaptive to the noisy, resource-limited nature of today’s quantum hardware. To that end, it integrates established error-mitigation methods — such as zero-noise extrapolation and dynamical decoupling — directly into compiler infrastructures. These techniques will be packaged as modules that can automatically adjust and combine based on performance analysis, enabling compilers to explore large design spaces and produce optimized, low-noise quantum programs with minimal manual intervention. In parallel, this project also explores new approaches to analog quantum programming or quantum simulation, and has developed the programming language SimuQ which treats quantum Hamiltonian evolution as the central object.

We model repetitive quantum error correction (QEC) with the single-error-correcting five-qubit code on a network of individually controlled qubits with always-on Ising couplings. We use our previously designed universal set of quantum gates based on sequences of shaped decoupling pulses. In addition to being accurate quantum gates, the sequences also provide dynamical decoupling (DD) of low-frequency phase noise. The simulation involves integrating the unitary dynamics of six qubits over the duration of tens of thousands of control pulses, using classical stochastic phase noise as a source of decoherence. The combined DD and QEC protocol dramatically improves the coherence, with the QEC alone being responsible for more than an order of magnitude infidelity reduction.

Quantum error correcting code (QECC) with encoding and decoding circuits having low gate count, is an important criterion for realizing quantum computing systems. CSS QECCs are known to have simple circuits. Shaw et al. showed that it is not possible to have a 6-qubit CSS type QECC without sharing entanglement between the encoder and the decoder. In this paper, we propose a 6-qutrit approximate QECC (AQECC) of CSS structure which can simultaneously correct phase errors in upto six qutrits, and one bit error in only four of the six qutrits, without sharing prior entanglement. Our AQECC corrects a single error with probability 0.75 for symmetric error model, and probability 0.9988 for asymmetric error model. It also maintains CSS structure without sharing prior entanglement. Furthermore, the quantum cost of a circuit for this AQECC is 55.72% less than that for the 9-qutrit exact QECC. Overall, low qutrit count, low cost circuit realization with low depth and the ability to correct multiple phase errors make our proposed AQECC a suitable candidate for real life quantum channels.

The quantum IO monad is a purely functional interface to quantum programming implemented as a Haskell library. At the same time it provides a constructive semantics of quantum programming. The QIO monad separates reversible (i.e., unitary) and irreversible (i.e., probabilistic) computations and provides a reversible let operation (ulet), allowing us to use ancillas (auxiliary qubits) in a modular fashion. QIO programs can be simulated either by calculating a probability distribution or by embedding it into the IO monad using the random number generator. As an example we present a complete implementation of Shor's algorithm.

In this thesis, a jointed preamble/boundary detection and fractional CFO estimation design is presented which supports dual SC/HSI modes of IEEE 802.15.3c applications. Based on correlation based algorithms which utilizes the structure of preamble, an efficiency architecture is proposed which realizes synchronization detection with a sequential detection scheme and a single hardware for dual SC/HSI modes and three detection operations. In order to achieve the requirement of sampling frequency, the baseband design is 8x parallelism and is operating at 333 MHz clock rate. The achieved bit error rate (uncoded) at 12 dB SNR for SC and HSI modes are 8.92×10-4 and 1.43×10-2, respectively. The proposed architecture of synchronization detection is implemented using 65 nm 1P9M CMOS process and the total gate count is 189k with power consumption of 60.16 mW including memory elements which occupies 63.26 % and can be shared with the frequency domain equalizer. Moreover, the power consumption is saved 42.5 % by applying low power techniques.