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.
We investigate the counterparts of random walk in universal quantum computing and their implementation using standard quantum circuits. Quantum walk have been recently well investigated for traversing graphs with certain oracles. We focus our study on traversing a 1-D graph, namely a circle, and show how to implement discrete circular quantum walk in quantum circuits built with universal CNOT and single quit gates. We review elementary quantum gates and circuit decomposition and propose a a generalized version of the all CNOT based quantum discrete circular walk. We simulated these circuits on…
We show how to efficiently simulate continuous-time quantum query algorithms that run in time T in a manner that preserves the query complexity (within a polylogarithmic factor) while also incurring a small overhead cost in the total number of gates between queries. By small overhead, we mean T within a factor that is polylogarithmic in terms of T and a cost measure that reflects the cost of computing the driving Hamiltonian. This permits any continuous-time quantum algorithm based on an efficiently computable driving Hamiltonian to be converted into a gate-efficient algorithm with similar run…
Equivalence of continuous- and discrete-variable gate-based quantum computers with finite energy
Chalmers University of Technology（查尔姆斯理工大学）；Technical University of Denmark（丹麦技术大学）；Johannes-Gutenberg University of Mainz（约翰内斯·古腾堡美因茨大学）；University of Oxford（牛津大学）
We construct a primitive gate set for the digital quantum simulation of the binary tetrahedral ($\mathbb{BT}$) group on two quantum architectures. This nonabelian discrete group serves as a crude approximation to $SU(2)$ lattice gauge theory while requiring five qubits or one quicosotetrit per gauge link. The necessary basic primitives are the inversion gate, the group multiplication gate, the trace gate, and the $\mathbb{BT}$ Fourier transform over $\mathbb{BT}$. We experimentally benchmark the inversion and trace gates on ibm nairobi, with estimated fidelities between $14-55\%$, depending on…
We present a simple method for constructing optimal fault-tolerant approximations of arbitrary unitary gates using an arbitrary discrete universal gate set. The method presented is numerical and scales exponentially with the number of gates used in the approximation. However, for the specific case of arbitrary single-qubit gates and the fault-tolerant gates permitted by the concatenated 7-qubit Steane code, we find gate sequences sufficiently long and accurate to permit the fault-tolerant factoring of numbers thousands of bits long. A general scaling law of how rapidly these fault-tolerant app…
We show that qubit stabilizer states can be represented by non-negative quasi-probability distributions associated with a Wigner-Weyl-Moyal formalism where Clifford gates are positive state-independent maps. This is accomplished by generalizing the Wigner-Weyl-Moyal formalism to three generators instead of two---producing an exterior, or Grassmann, algebra---which results in Clifford group gates for qubits that act as a permutation on the finite Weyl phase space points naturally associated with stabilizer states. As a result, a non-negative probability distribution can be associated with each …
The goal of this paper is to introduce building blocks for adiabatic quantum algorithms. Adiabatic quantum computing uses the principle of quantum annealing, which implies that a carefully controlled energy solution is optimal and corresponds to minimizing a discrete function. The input function can be influenced by rewards and penalties to favor a solution that meets restrictions that are imposed by the problem. We show how to accomplish this influence for gates in adiabatic quantum computing, particularly the controlled-NOT gate (CNOT gate) which is fundamental to all quantum gates on two or…
Quantum information processing with hybrid protocols making use of discrete- and continuous-variable currently attracts of great interest because of its promising applications in scalable quantum computer and distant quantum network. By inducing a giant cross-Kerr nonlinearity between two cavities, we propose a general protocol for hybrid quantum gate and quantum entanglement with high fidelity between a stationary, discrete photonic qubit and a flying photonic state. Interestingly, our protocol can be used to conduct a controlled-Z quantum gate between a stationary microwave photon stored in …
Quantum walk has been regarded as a primitive to universal quantum computation. By using the operations required to describe the single particle discrete-time quantum walk on a position space we demonstrate the realization of the universal set of quantum gates on two- and three-qubit systems. The idea is to utilize the effective Hilbert space of the single qubit and the position space on which it evolves in order to realize multi-qubit states and universal set of quantum gates on them. Realization of many non-trivial gates and engineering arbitrary states is simpler in the proposed quantum wal…
One of the lowest-order corrections to Gaussian quantum mechanics in infinite-dimensional Hilbert spaces are Airy functions: a uniformization of the stationary phase method applied in the path integral perspective. We introduce a "periodized stationary phase method" to discrete Wigner functions of systems with odd prime dimension and show that the $\fracπ{8}$ gate is the discrete analog of the Airy function. We then establish a relationship between the stabilizer rank of states and the number of quadratic Gauss sums necessary in the periodized stationary phase method. This allows us to develop…
We apply the methodology of optimal control theory to the problem of implementing quantum gates in continuous variable systems with quadratic Hamiltonians. We demonstrate that it is possible to define a fidelity measure for continuous variable (CV) gate optimization that is devoid of traps, such that the search for optimal control fields using local algorithms will not be hindered. The optimal control of several quantum computing gates, as well as that of algorithms composed of these primitives, is investigated using several typical physical models and compared for discrete and continuous quan…
The discrete Fourier transform (DFT) is of fundamental interest in photonic quantum information, yet the ability to scale it to high dimensions depends heavily on the physical encoding, with practical recipes lacking in emerging platforms such as frequency bins. In this Letter, we show that d-point frequency-bin DFTs can be realized with a fixed three-component quantum frequency processor (QFP), simply by adding to the electro-optic modulation signals one radio-frequency harmonic per each incremental increase in d. We verify gate fidelity F > 0.9997 and success probability P > 0.965 up to d = …
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.
For a quantum computer acting on d-dimensional systems, we analyze the computational power of circuits wherein stabilizer operations are perfect and we allow access to imperfect non-stabilizer states or operations. If the noise rate affecting the non-stabilizer resource is sufficiently high, then these states and operations can become simulable in the sense of the Gottesman-Knill theorem, reducing the overall power of the circuit to no better than classical. In this paper we find the depolarizing noise rate at which this happens, and consequently the most robust non-stabilizer states and non-C…
The controlled-NOT gate and controlled square-root NOT gate play an important role in quantum algorithm. This article reports the experimental results of these two universal quantum logic gates (controlled square-root NOT gate and controlled-NOT gate) on a 7-qubit NMR quantum computer. Further, we propose a simple experimental method to measure and correct the error in the controlled phase-shift gate, which is helpful to construct a more perfect phase-shift gate experimentally and can also be used in more qubits discrete Fourier transformation.
In this paper we study the properties of two-qubit gates. We review the most common parameterizations for the local equivalence classes of two-qubit gates and the connections between them. We then introduce a new discrete local invariant, namely the number of local degrees of freedom that a gate can bind. The value of this invariant is calculated analytically for all the local equivalence classes of two-qubit gates. We find that almost all two-qubit gates can bind the full six local degrees of freedom and are in this sense more effective than the controlled-NOT gate, which only can bind four l…
Among the many proposals for the realization of a quantum computer, holonomic quantum computation (HQC) is distinguished from the rest in that it is geometrical in nature and thus expected to be robust against decoherence. Here we analyze the realization of various quantum gates by solving the inverse problem: Given a unitary matrix, we develop a formalism by which we find loops in the parameter space generating this matrix as a holonomy. We demonstrate for the first time that such a one-qubit gate as the Hadamard gate and such two-qubit gates as the CNOT gate, the SWAP gate and the discrete F…
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…
We present a formula for an infinite number of universal quantum logic gates, which are $4$ by $4$ unitary solutions to the Yang-Baxter (Y-B) equation. We obtain this family from a certain representation of the cyclic group of order $n$. We then show that this {\it discrete} family, parametrized by integers $n$, is in fact, a small sub-class of a larger {\it continuous} family, parametrized by real numbers $θ$, of universal quantum gates. We discuss the corresponding Yang-Baxterization and related symmetries in the concomitant Hamiltonian.
In this work, we report on a novel quantum gate approximation algorithm based on the application of parametric two-qubit gates in the synthesis process. The utilization of these parametric two-qubit gates in the circuit design allows us to transform the discrete combinatorial problem of circuit synthesis into an optimization problem over continuous variables. The circuit is then compressed by a sequential removal of two-qubit gates from the design, while the remaining building blocks are continuously adapted to the reduced gate structure by iterated learning cycles. We implemented the develope…
We propose a variational quantum eigensolver (VQE) algorithm that uses a fault-tolerant gate-set, and is hence suitable for implementation on a future error-corrected quantum computer. VQE quantum circuits are typically designed for near-term, noisy quantum devices and have continuously parameterized rotation gates as the central building block. On the other hand, a fault-tolerant quantum computer can only implement a discrete set of logical gates, such as the so-called Clifford+T gates. We show that the energy minimization of VQE can be performed with such a fault-tolerant discrete gate-set, …
Essentially all known quantum gates rely on a weak-coupling approximation resulting in linear dynamics. With the explicit example of trapped ions, we show how high-fidelity quantum gates can be achieved outside such an approximation, and we derive readily implementable driving fields to realize gates with extremely high fidelities for ions well outside the Lamb-Dicke regime with motional temperatures achievable by only Doppler cooling.
The sensitivity of quantum systems to external disturbances is a fundamental problem for the implementation of functional quantum devices, quantum information and computation. Based on remarkable experimental progress in optics and ultra-cold gases, we study the consequences of a short-time (instantaneous) noise while an intensity-dependent phase acquisition is associated with a qubit propagating on $N$-cycle. By employing quantum coherence measures, we report emerging unstable regimes in which hitherto unknown quantum walks arise, such as self-focusing and breathing dynamics. Our results unve…
We present a scheme for non-deterministically approximating photon number resolving detectors using non-discriminating detectors. The model is simple in construction and employs very few physical resources. Despite its non-determinism, the proposal may nonetheless be suitable for use in some quantum optics experiments in which non-determinism can be tolerated. We analyze the detection scheme in the context of an optical implementation of the controlled-NOT gate, an inherently non-deterministic device. This allows the gate's success probability to be traded away for improved gate fidelity, assu…
The accurate implementation of quantum gates is essential for the realisation of quantum algorithms and digital quantum simulations. This accuracy may be increased on noisy hardware through the variational optimisation of gates, however the experimental realisation of such a protocol is impeded by the large effort required to estimate the fidelity of an implemented gate. With a hierarchy of approximations we find a faithful approximation to the quantum process fidelity that can be estimated experimentally with reduced effort. Its practical use is demonstrated with the optimisation of a three-q…
In the past few years there has been a tumultuous activity aimed at introducing novel conceptual schemes for quantum computing. The approach proposed in (Marzuoli A and Rasetti M 2002, 2005a) relies on the (re)coupling theory of SU(2) angular momenta and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum-circuit model based on `qubits' and Boolean gates. Computational states belong to finite-dimensional Hilbert spaces labelled by both discrete and continuous parameters, and unitary gates may depend on quantum numbers ranging over finite sets of val…
We examine the validity of the rotating wave approximation (RWA) in non-adiabatic holonomic single-qubit gates [New J. Phys. {\bf 14}, 103035 (2012)]. We demonstrate that the adoption of RWA may lead to a sharp decline in fidelity for rapid gate implementation and small energy separation between the excited and computational states. The validity of the RWA in the recent experimental realization [Nature (London) {\bf 496}, 482 (2013)] of non-adiabatic holonomic quantum computation for a superconducting qubit is examined.
A bound on the error introduced by truncating a quantum addition is given. This bound shows that only a few controlled rotation gates will be necessary to get a reliable computation.
The isoholonomic problem in a homogeneous bundle is formulated and solved exactly. The problem takes a form of a boundary value problem of a variational equation. The solution is applied to the optimal control problem in holonomic quantum computer. We provide a prescription to construct an optimal controller for an arbitrary unitary gate and apply it to a $ k $-dimensional unitary gate which operates on an $ N $-dimensional Hilbert space with $ N \geq 2k $. Our construction is applied to several important unitary gates such as the Hadamard gate, the CNOT gate, and the two-qubit discrete Fourie…
We propose a novel scheme to implement a quantum controlled phase gate for trapped ions in thermal motion with one standing wave laser pulse. Instead of applying the rotating wave approximation this scheme makes use of the counter-rotating terms of operators. We also demonstrate that the same scheme can be used to generate maximally entangled states of $N$ trapped ions by a single laser pulse.
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.
In this Letter, we present a physical scheme for implementing the discrete quantum Fourier transform in a coupled semiconductor double quantum dot system. The main controlled-R gate operation can be decomposed into many simple and feasible unitary transformations. The current scheme would be a useful step towards the realization of complex quantum algorithms in the quantum dot system.
The hybrid approach to quantum computation simultaneously utilizes both discrete and continuous variables which offers the advantage of higher density encoding and processing powers for the same physical resources. Trapped ions, with discrete internal states and motional modes which can be described by continuous variables in an infinite dimensional Hilbert space, offer a natural platform for this approach. A nonlinear gate for universal quantum computing can be implemented with the conditional beam splitter Hamiltonian $|e\rangle \langle e| ( a^{\dagger} b + a b^{\dagger})$ that swaps the qua…
Noisy Intermediate-Scale Quantum (NISQ) devices fail to produce outputs with sufficient fidelity for deep circuits with many gates today. Such devices suffer from read-out, multi-qubit gate and crosstalk noise combined with short decoherence times limiting circuit depth. This work develops a methodology to generate shorter circuits with fewer multi-qubit gates whose unitary transformations approximate the original reference one. It explores the benefit of such generated approximations under NISQ devices. Experimental results with Grover's algorithm, multiple-control Toffoli gates, and the Tran…
We present an algorithm for building a circuit that approximates single qubit unitaries with precision ε using O(log(1/ε)) Clifford and T gates and employing up to two ancillary qubits. The algorithm for computing our approximating circuit requires an average of O(log^2(1/ε)log log(1/ε)) operations. We prove that the number of gates in our circuit saturates the lower bound on the number of gates required in the scenario when a constant number of ancillae are supplied, and as such, our circuits are asymptotically optimal. This results in significant improvement over the current state of the art…
Addressability of spin qubits in a silicon double quantum dot setup in the (1,1) charge configuration relies on having a large difference between the Zeeman splittings of the electrons. When the difference is not sufficiently large, the rotating wave approximation becomes inaccurate. We consider a device working in this regime, with always-on exchange coupling, and describe how a CZ gate and arbitrary one-qubit gates which are robust against charge noise can be implemented by smoothly pulsing the microwave source, while eliminating the crosstalk. We find that the most significant deviations fr…
We present a theoretical study of single-qubit operations by oscillatory fields on various semiconductor platforms. We explicitly show how to perform faster gate operations by going beyond the universally-used rotating wave approximation (RWA) regime, while using only two sinusoidal pulses. No complicated pulse shaping or optimal control sequences are required. We first show for specific published experiments how much error is currently incurred by implementing pulses designed using standard RWA. We then show that an even modest increase in gate speed would cause problems in using RWA for gate…
We present QEst, a procedure to systematically generate approximations for quantum circuits to reduce their CNOT gate count. Our approach employs circuit partitioning for scalability with procedures to 1) reduce circuit length using approximate synthesis, 2) improve fidelity by running circuits that represent key samples in the approximation space, and 3) reason about approximation upper bound. Our evaluation results indicate that our approach of "dissimilar" approximations provides close fidelity to the original circuit. Overall, the results indicate that QEst can reduce CNOT gate count by 30…
