Quantum computation may well be performed with the use of electric circuits. Especially, the Schrödinger equation can be simulated by the lumped-element model of transmission lines, which is applicable to low-frequency electric circuits. In this paper, we show that the Dirac equation is simulated by the distributed-element model, which is applicable to high-frequency electric circuits. Then, a set of universal quantum gates (the Hadamard, phase-shift and CNOT gates) are constructed by networks made of transmission lines. We demonstrate Shor's prime factorization based on electric circuits. It will be possible to simulate any quantum algorithms simply by designing networks of metallic wires.
The block canonical coset decomposition is developed as a universal algorithmic tool to synthesize n-qubit quantum gates out of experimentally realizable NMR elements. The two-, three-, and four-qubit quantum Fourier transformations are worked out as examples. The proposed decomposition bridges the state of the art numerical analysis with NMR quantum gate synthesis.
The effects of imperfect gate operations in implementation of Shor's prime factorization algorithm are investigated. The gate imperfections may be classified into three categories: the systematic error, the random error, and the one with combined errors. It is found that Shor's algorithm is robust against the systematic errors but is vulnerable to the random errors. Error threshold is given to the algorithm for a given number $N$ to be factorized.
The effects of imperfect gate operations in implementation of Shor's prime factorization algorithm are investigated. The gate imperfections may be classified into three categories: the systematic error, the random error, and the one with combined errors. It is found that Shor's algorithm is robust against the systematic errors but is vulnerable to the random errors. Error threshold is given to the algorithm for a given number $N$ to be factorized.
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 …
Quantum integer factorization is a potential quantum computing solution that may revolutionize cryptography. Nevertheless, a scalable and efficient quantum algorithm for noisy intermediate-scale quantum computers looks far-fetched. We propose an alternative factorization method, within the digitized-adiabatic quantum computing paradigm, by digitizing an adiabatic quantum factorization algorithm enhanced by shortcuts to adiabaticity techniques. We find that this fast factorization algorithm is suitable for available gate-based quantum computers. We test our quantum algorithm in an IBM quantum c…
The non-adiabatic geometric quantum computation (NGQC) has attracted a lot of attention for noise-resilient quantum control. However, previous implementations of NGQC require long evolution paths that make them more vulnerable to incoherent errors than their dynamical counterparts.In this work, we experimentally realize a universal short-path non-adiabatic geometric gate set (SPNGQC) with a 2-times shorter evolution path on a superconducting quantum processor. Characterizing with both quantum process tomography and randomized benchmarking methods, we report an average single-qubit gate fidelit…
Universal quantum computation may be realized based on quantum walk, by formulating it as a scattering problem on a graph. In this paper, we simulate quantum gates through electric circuits, following a recent report that a one-dimensional $LC$ electric circuit can simulate a Schrödinger equation and hence a quantum walk. Especially, we propose a physical realization of a set of universal quantum gates consisting of the CNOT, Hadamard and $π/4$ phase-shift gates with the use of telegrapher wires and mixing bridges. Furthermore, we construct the $π/2^{n}$ phase-shift gate for an arbitrary integ…
For a one-dimensional chain of four nuclear spins (1/2) and taking into account first and second neighbor interactions among the spin system, we make the numerical simulation of Shor prime factorization algorithm of the integer number N=4 to study the influence of the second neighbor interaction on the performance of this algorithm. It is shown that the optimum Rabi's frequency to control the non-resonant effects is dominated by the second neighbor interaction coupling parameter ($J'$), and that a good Shor quantum factorization is achieved for a ratio of second to first coupling constant of $…
Quantum-Kit is a graphical desktop application for quantum circuit simulations. Its powerful, memory-efficient computational engine enables large-scale simulations on a desktop. The ability to design hybrid circuits, with both quantum and classical bits and controls, is employed to demonstrate Kitaev's approach to Shor's factorization algorithm. For the first time, Shor's factorization of a 24-bit integer is simulated with Quantum-Kit in a mere 26 minutes on a modest desktop with Intel Core i5 7400T, 2.4GHz and12GB RAM. While the largest number factorized so far has been a 20-bit integer, requ…
We propose an alternative scheme of shortcuts to quantum phase gate in a much shorter time based on the approach of Lewis-Riesenfeld invariants in cavity quantum electronic dynamics (QED) systems. This scheme can be used to perform one-qubit phase gate, two-qubit controlled phase gate and also multiqubit controlled phase gate. The strict numerical simulation for some quantum gates are given, and demonstrate that the total operation time of our scheme is shorter than previous schemes and very robustness against decoherence.
Shor's quantum algorithm is very important for cryptography, since it can factor large numbers much faster than classical algorithms. In this study, we implement a simulator for Shor's quantum algorithm on graphic processor units (GPU) and compare our results with Liquid -which is Microsoft quantum simulation platform- and two classical CPU-implementations. We evaluate 10 benchmarks for comparing our GPU implementation with Liquid and single-core implementation. The analysis shows that GPU vector operations is more suitable for Shor's quantum algorithm. Our GPU kernel function is compute-bound…
It is commonly assumed that Shor's quantum algorithm for the efficient factorization of a large number $N$ requires a pure initial state. Here we demonstrate that a single pure qubit together with a collection of $log_2 N$ qubits in an arbitrary mixed state is sufficient to implement Shor's factorization algorithm efficiently.
We present a novel and efficient in terms of circuit depth design for Shor's quantum factorization algorithm. The circuit effectively utilizes a diverse set of adders based on the quantum Fourier transform (QFT) Draper's adders to build more complex arithmetic blocks: quantum multiplier/accumulators by constants and quantum dividers by constants. These arithmetic blocks are effectively architected into a generic modular quantum multiplier which is the fundamental block for modular exponentiation circuit, the most computational intensive part of Shor's algorithm. The proposed modular exponentia…
We theoretically propose an alternative method to realize a deterministic SWAP gate using shortcuts to adiabatic passage based on the approach of Lewis- Riesenfeld invariants in cavity quantum electronic dynamics (QED). By combining Lewis-Riesenfeld invariants with quantum Zeno dynamics, the SWAP gate can be achieved deterministically. The strict numerical results show that our scheme is a fast and robust approach to achieve SWAP gate.
We propose a method for applying the quantum error-correction method for errors that occur during quantum gates. Using a perturbation treatment of the noise that allows us to separate it from the ideal evolution of the quantum gate, we demonstrate that in certain cases it is necessary to divide the quantum gate in short time steps intercalated by correction procedures. A prescription of how these gates can be constructed is provided, as well as a proof that, even for the cases when the division of the quantum gate in short time steps is not necessary, this method may be advantageous for reduci…
We investigate how to attack Shor's quantum algorithm for factorization with an entangling probe. We show that an attacker can steal an exact solution of Shor's algorithm outside an institute where the quantum computer is installed if he replaces its initialized quantum register with entangled qubits, namely the entangling probe. He can apply arbitrary local operations to his own probe. Moreover, we assume that there is an unauthorized person who helps the attacker to commit a crime inside the institute. He tells garbage data obtained from measurements of the quantum register to the attacker s…
Fault-tolerant quantum computers compose elements of a discrete gate set in order to approximate a target unitary. The problem of minimising the number of gates is known as gate-synthesis. The approximation error is a form of coherent noise, which can be significantly more damaging than comparable incoherent noise. We show how mixing over different gate sequences can convert this coherent noise into an incoherent form. As measured by diamond distance, the post-mixing noise is quadratically smaller than before mixing, with no additional resource cost. Equivalently, we can use a shorter gate seq…
Fault-tolerant quantum computation using quantum error-correcting codes requires fault-tolerant constructions of nontransversal gates. Shor proposed a fault-tolerant construction of a nontransversal gate, i.e., the Toffoli gate for a family of the Calderbank-Shor-Steane codes which has even-weight stabilizer generators and odd-weight logical operators. In fact, as shown in this paper, error propagations in the ancilla preparation of Shor's construction lead to logical errors. This work provides a solution to this problem through the insertion of bit flip error correction. Finally, error rates …
We study effects of imperfections induced by residual couplings between qubits on the accuracy of Shor's algorithm using numerical simulations of realistic quantum computations with up to 30 qubits. The factoring of numbers up to N=943 show that the width of peaks, which frequencies allow to determine the factors, grow exponentially with the number of qubits. However, the algorithm remains operational up to a critical coupling strength $ε_c$ which drops only polynomially with $\log_2 N$. The numerical dependence of $ε_c$ on $\log_2 N$ is explained by analytical estimates that allows to obtain …
Nonadiabatic holonomic quantum computation has attracted increasing interest because of its robustness. In this paper, based on the shortcuts to adiabaticity (STA), we propose a scheme to construct a three-qubit nonadiabatic holonomic gate in a cavity quantum electrodynamics system. In the scheme, the three-qubit holonomic gate is implemented in a single step. Besides, numerical simulations show that the constructed holonomic quantum gate holds the characteristics of high-fidelity, decoherence-resistance and error-immunity.
Nonadiabatic geometric quantum computation is dedicated to the realization of high-fidelity and robust quantum gates, which are necessary for fault-tolerant quantum computation. However, it is limited by cyclic and mutative evolution path, which usually requires longer gate-time and abrupt pulse control, weakening the gate performance. Here, we propose a scheme to realize geometric quantum gates with noncyclic and nonadiabatic evolution via invariant-based shortcuts, where universal quantum gates can be induced in one step without path mutation and the gate time is also effectively shortened. …
We give algorithms to factorize large integers in the duality computer. We provide three duality algorithms for factorization based on a naive factorization method, the Shor algorithm in quantum computing, and the Fermat's method in classical computing. All these algorithms are polynomial in the input size.
In a typical quantum circuit, nonlocal quantum gates are applied to nonproximal qubits. If the underlying physical interactions are short-range (e.g., exchange interactions between spins), intermediate swap operations must be introduced, thus increasing the circuit depth. Here we develop a class of "mediated" gates for spin qubits, which act on nonproximal spins via intermediate ancilla qubits. At the end of the operation, the ancillae return to their initial states. We show how these mediated gates can be used (1) to generate arbitrary quantum states and (2) to construct arbitrary quantum gat…
Based on a `shortcut-to-adiabaticity' (STA) scheme, we theoretically design and experimentally realize a set of high-fidelity single-qubit quantum gates in a superconducting Xmon qubit system. Through a precise microwave control, the qubit is driven to follow a fast `adiabatic' trajectory with the assistance of a counter-diabatic field and the correction of derivative removal by adiabatic gates. The experimental measurements of quantum process tomography and interleaved randomized benchmarking show that the process fidelities of our STA quantum gates are higher than 94.9% and the gate fideliti…
For a frequency-tunable two-qubit system, a controlled-Z (CZ) gate can be realized by adiabatically driving the qubit system through an avoided level crossing between an auxiliary state and computational levels. Here, we theoretically propose a fast CZ gate using a shortcut-to-adiabaticity (STA). Experimentally, the STA CZ gate is implemented with a 52 ns control pulse for two coupled superconducting Xmon qubits. Measured fidelity of the STA CZ gate is higher than 96.0%, in both quantum process tomography and randomized benchmarking. The protocol allows a flexible design of the evolution time …
We introduce a shortcut to the adiabatic gate teleportation model of quantum computation. More specifically, we determine fast local counterdiabatic Hamiltonians able to implement teleportation as a universal computational primitive. In this scenario, we provide the counterdiabatic driving for arbitrary n-qubit gates, which allows to achieve universality through a variety of gate sets. Remarkably, our approach maps the superadiabatic Hamiltonian for an arbitrary n-qubit gate teleportation into the implementation of a rotated superadiabatic dynamics of an n-qubit state teleportation. This resul…
We formulate and numerically simulate the single control qubit Shor algorithm for the case of static imperfections induced by residual couplings between qubits. This allows us to study the accuracy of Shor's algorithm with respect to these imperfections using numerical simulations of realistic quantum computations with up to $n_q=18$ computational qubits allowing to factor numbers up to N=205193. We confirm that the algorithm remains operational up to a critical coupling strength $ε_c$ which drops only polynomially with $\log_2 N$. The obtained numerical dependence of $ε_c$ on $\log_2 N$ is in…
Geometric phases are robust against certain types of local noises, and thus provide a promising way towards high-fidelity quantum gates. However, comparing with the dynamical ones, previous implementations of nonadiabatic geometric quantum gates usually require longer evolution time, due to the needed longer evolution path. Here, we propose a scheme to realize nonadiabatic geometric quantum gates with short paths based on simple pulse control techniques, instead of deliberated pulse control in previous investigations, which can thus further suppress the influence from the environment induced n…
Nonadiabatic holonomic quantum computation uses non-Abelian geometric phases to implement a universal set of quantum gates that are robust against control imperfections and decoherence. Until now, a number of three-level-based schemes of nonadiabatic holonomic computation have been put forward, and several of them have been experimentally realized. However, all these works are based on the same class of nonadiabatic paths, which originates from the first nonadiabatic holonomic proposal. Here, we propose a universal set of nonadiabatic holonomic gates based on an extended class of nonadiabatic …
We investigate the time T a quantum computer requires to factorize a given number dependent on the number of bits L required to represent this number. We stress the fact that in most cases one has to take into account that the execution time of a single quantum gate is related to the decoherence time of the qubits that are involved in the computation. Although exhibited here only for special systems, this inter-dependence of decoherence and computation time seems to be a restriction in many current models for quantum computers and leads to the result that the computation time T scales much str…
Composite pulses, originally developed in Nuclear Magnetic Resonance (NMR), have found widespread use in experimental quantum information processing (QIP) to reduce the effects of systematic errors. Most pulses used so far have simply been adapted from existing NMR designs, and while techniques have been developed for designing composite pulses with arbitrary precision the results have been quite complicated and have found little application. Here I describe techniques for designing short but effective composite pulses to implement robust NOT gates, bringing together existing insights from NMR…
High-fidelity quantum gates are essential for large-scale quantum computation. However, any quantum manipulation will inevitably affected by noises, systematic errors and decoherence effects, which lead to infidelity of a target quantum task. Therefore, implementing high-fidelity, robust and fast quantum gates is highly desired. Here, we propose a fast and robust scheme to construct high-fidelity holonomic quantum gates for universal quantum computation based on resonant interaction of three-level quantum systems via shortcuts to adiabaticity. In our proposal, the target Hamiltonian to induce …
We analyse the design and optimisation of quantum logic circuits suitable for the experimental demonstration of a three-qubit quantum computation prototype based on optically-controlled, solid-state quantum logic gates. In these gates, the interaction between two qubits carried by the electron-spin of donors is mediated by the optical excitation of a control particle placed in their proximity. First, we use a geometrical approach for analysing the entangling characteristics of these quantum gates. Then, using a genetic programming algorithm, we develop circuits for the refined Deutsch-Jozsa al…
Typical circuit implementations of Shor's algorithm involve controlled rotation gates of magnitude $π/2^{2L}$ where $L$ is the binary length of the integer N to be factored. Such gates cannot be implemented exactly using existing fault-tolerant techniques. Approximating a given controlled $π/2^{d}$ rotation gate to within $δ=O(1/2^{d})$ currently requires both a number of qubits and number of fault-tolerant gates that grows polynomially with $d$. In this paper we show that this additional growth in space and time complexity would severely limit the applicability of Shor's algorithm to large in…
Nonadiabatic holonomic quantum computation (NHQC) has attracted significant attention due to its fast evolution and the geometric nature induced resilience to local noises. However, its long operation time and complex physical implementation make it hard to surpass the dynamical scheme, and thus hindering its wide application. Here, we present to implement NHQC with the shortest path under some conditions, through the inverse Hamiltonian engineering technique, which posseses higher fidelity and stronger robustness than previous NHQC schemes. Meanwhile, the gate performance in our scheme can be…
Fast and robust quantum gates is the cornerstone of fault-tolerance quantum computation. In this paper, we propose to achieve quantum gates based on non-cyclic geometric evolution. Dynamical phase during the evolution is cancelled by spin-echo process and the adiabatic control can be sped up through shortcut to adiabatic manner. Different from geometric gates based on cyclic evolution, the superiority of non-cyclic scheme is that the operation time is proportional to the rotation angle (but not the geometric phase) of quantum state. Therefore, the non-cyclic scheme becomes fairly fast in the c…
The complexity of a quantum gate, defined as the minimal number of elementary gates to build it, is an important concept in quantum information and computation. It is shown recently that the complexity of quantum gates built from random quantum circuits almost surely grows linearly with the number of building blocks. In this article, we provide two short proofs of this fact. We also discuss a discrete version of quantum circuit complexity growth.
We present a fault-tolerant universal gate set consisting of Hadamard and controlled-controlled-Z (CCZ) on Bacon-Shor subsystem codes. Transversal non-Clifford gates on these codes are intriguing in that higher levels of the Clifford hierarchy become accessible as the code becomes more asymmetric. For instance, in an appropriate gauge, Bacon-Shor codes on an $m\times m^k$ lattice have transversal $k$-qubit-controlled $Z$. Through a variety of tricks, including intermediate error-correction and non-Pauli recovery, we reduce the overhead required for fault-tolerant CCZ. We calculate pseudothresh…
In the present work the extrema of the objective functional for the problem of generation of quantum gates (logical elements for quantum computations) for two--level systems are investigated for short duration of the control. The problem of existence of local but not global extrema, the so called traps, is considered. In prior works the absence of traps was proved for a sufficiently long control duration. In this paper we prove that for almost all target unitary operators and system Hamiltonians traps are absent for an arbitrarily small control duration. For the remainder target unitary operat…
We propose an adiabatic quantum algorithm capable of factorizing numbers, using fewer qubits than Shor's algorithm. We implement the algorithm in an NMR quantum information processor and experimentally factorize the number 21. Numerical simulations indicate that the running time grows only quadratically with the number of qubits.
Inspired by non-abelian vortex anyons in spinor Bose-Einstein condensates, we consider the quantum double $\mathcal{D}(Q_8)$ anyon model as a platform to carry out a particular instance of Shor's factorization algorithm. We provide the excitation spectrum, the fusion rules, and the braid group representation for this model, and design a circuit architecture that facilitates the computation. All necessary quantum gates, less one, can be compiled exactly for this hybrid topological quantum computer, and to achieve universality the last operation can be implemented in a non-topological fashion. T…
We try to minimize the number of qubits needed to factor an integer of n bits using Shor's algorithm on a quantum computer. We introduce a circuit which uses 2n+3 qubits and O(n^3 lg(n)) elementary quantum gates in a depth of O(n^3) to implement the factorization algorithm. The circuit is computable in polynomial time on a classical computer and is completely general as it does not rely on any property of the number to be factored. Keywords: Factorization, quantum circuits, modular arithmetics
The Non-Identity Check problem asks whether a given a quantum circuit is far away from the identity or not. It is well known that this problem is QMA-Complete \cite{JWB05}. In this note, it is shown that the Non-Identity Check problem remains QMA-Complete for circuits of short depth. Specifically, we prove that for constant depth quantum circuit in which each gate is given to at least $Ω(\log n)$ bits of precision, the Non-Identity Check problem is QMA-Complete. It also follows that the hardness of the problem remains for polylogarithmic depth circuit consisting of only gates from any universa…
The road to computing on quantum devices has been accelerated by the promises that come from using Shor's algorithm to reduce the complexity of prime factorization. However, this promise hast not yet been realized due to noisy qubits and lack of robust error correction schemes. Here we explore a promising, alternative method for prime factorization that uses well-established techniques from variational imaginary time evolution. We create a Hamiltonian whose ground state encodes the solution to the problem and use variational techniques to evolve a state iteratively towards these prime factors.…
Quantum computing technology may soon deliver revolutionary improvements in algorithmic performance, but these are only useful if computed answers are correct. While hardware-level decoherence errors have garnered significant attention, a less recognized obstacle to correctness is that of human programming errors -- "bugs". Techniques familiar to most programmers from the classical domain for avoiding, discovering, and diagnosing bugs do not easily transfer, at scale, to the quantum domain because of its unique characteristics. To address this problem, we have been working to adapt formal meth…
We consider a probabilistic quantum implementation of a variable of the Pocklington-Lehmer $N-1$ primality test using Shor's algorithm. O($\log^3 N \log\log N \log\log\log N$) elementary q-bit operations are required to determine the primality of a number $N$, making it (asymptotically) the fastest known primality test. Thus, the potential power of quantum mechanical computers is once again revealed.
