Based on a quantum analysis of two capacitively coupled current-biased Josephson junctions, we propose two fundamental two-qubit quantum logic gates. Each of these gates, when supplemented by single-qubit operations, is sufficient for universal quantum computation. Numerical solutions of the time-dependent Schroedinger equation demonstrate that these operations can be performed with good fidelity.
It is shown that certain natural quantum logic gates, {\it i.e.} unitary time evolution matrices for spin-\frac{1}{2} quantum spins, can be represented as sums, with appropriate phases, over classical logic gates, in a direct analogy with the Feynman path integral representation of quantum mechanics. On the other hand, it is shown that a natural quantum gate obtained by analytically continuing the transfer matrix of the anisotropic nearest-neighbour Ising model to imaginary time, does not admit such a representation.
A method for synthesizing quantum gates is presented based on interpolation methods applied to operators in Hilbert space. Starting from the diagonal forms of specific generating seed operators with non-degenerate eigenvalue spectrum one obtains for arity-one a complete family of logical operators corresponding to all the one-argument logical connectives. Scaling-up to n-arity gates is obtained by using the Kronecker product and unitary transformations. The quantum version of the Fourier transform of Boolean functions is presented and a Reed-Muller decomposition for quantum logical gates is de…
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…
We analyze analytically and numerically quantum logic gates in a one-dimensional spin chain with Heisenberg interaction. Analytic solutions for basic one-qubit gates and swap gate are obtained for a quantum computer based on logical qubits. We calculated the errors caused by imperfect pulses which implement the quantum logic gates. It is numerically demonstrated that the probability error is proportional to $ε^4$, while the phase error is proportional to $ε$, where $ε$ is the characteristic deviation from the perfect pulse duration.
This is a brief overview of quantum holonomies in the context of quantum computation. We choose an adequate set of quantum logic gates, namely, a phase gate, the Hadamard gate, and a conditional-phase gate and show how they can be implemented by purely geometric means. Such gates may be more resilient to certain types of errors.
We show that the quantum logic gates, {\it viz.} the single qubit Hadamard and Phase Shift gates, can also be realised using q-deformed angular momentum states constructed via the Jordan-Schwinger mechanism with two q-deformed oscillators. {\it Keywords :} quantum logic gates ; q-deformed oscillators ; quantum computation {\it PACS:} 03.67.Lx ; 02.20.Uw
Quantum algorithms often benefit from the ability to execute multi-qubit (>2) gates. To date such multi-qubit gates are typically decomposed into single- and two-qubit gates, particularly in superconducting qubit architectures. The ability to perform multi-qubit operations in a single step could vastly improve the fidelity and execution time of many algorithms. Here, we propose a single shot method for executing an i-Toffoli gate, a three-qubit gate gate with two control and one target qubit, using currently existing superconducting hardware. We show numerical evidence for a process fidelity o…
This is an exposition of some basic mathematical aspects of quantum logic gates. At first we established some general formulas for the case of arbitrary quantum gate A with unique restriction A^2=I. The explicit form of the generators and roots of matrix A have been found . Then we apply general results to the particular cases of one-qubit and multi-qubit quantum gates. Some interesting properties of square roots of basic Pauli and Hadamard gates are demonstrated.
We present a scheme to realise the basic two-quibit logic gates such as quantum phase gate and controlle-NOT gate using a detuned optical cavity interacting with a three-level Raman system. We discuss the role of Stark shifts which are as important as the terms leading to two-photon transition. The operation of the proposed logic gates involves metastable states of the atom and hence is not affected by spontaneous emission. These ideas can be extended to produce multiparticle entanglement.
Topologically quantum error corrected logical gates are complex. Chains of errors can form in space and time and diagonally in spacetime. It is highly nontrivial to determine whether a given logical gate is free of low weight combinations of errors leading to failure. We report a new tool Nestcheck capable of analyzing an arbitrary topological computation and determining the minimum number of errors required to cause failure.
The authors call attention to a previous work [Qing Lin and Bing He, Phys. Rev. A 80, 042310 (2009)] on the realization of multi-qubit logic gates with controlled-path and merging gate. We supplement the work by showing how to efficiently build realistic quantum circuits in this approach and giving the guiding rules for the task.
The implementation of fault-tolerant quantum gates on encoded logic qubits is considered. It is shown that transversal implementation of logic gates based on simple geometric control ideas is problematic for realistic physical systems suffering from imperfections such as qubit inhomogeneity or uncontrollable interactions between qubits. However, this problem can be overcome by formulating the task as an optimal control problem and designing efficient algorithms to solve it. In particular, we can find solutions that implement all of the elementary logic gates in a fixed amount of time with limi…
We apply residuated structures associated with fuzzy logic to develop certain aspects of information processing in quantum computing from a logical perspective. For this purpose, we introduce an axiomatic system whose natural interpretation is the irreversible quantum Poincare structure.
In this article the elementary gates for ternary quantum logic circuit are studied. We propose the ternary controlled X (TCX) gate or ternary controlled Z (TCZ) gate as two-qutrit elementary gate, which is universal when assisted by arbitrary one-qutrit gates. It is primitive, efficient and easy to implement. Based on Cartan decomposition, we also give the one-qutrit elementary gates. Then the synthesis of some important ternary gates is investigated and the scheme of physical implementation for these ternary gates is discussed. Finally we extend these elementary gates to more general qudit ca…
The basic principles and results of Conservative Logic introduced by Fredkin and Toffoli on the basis of a seminal paper of Landauer are extended to d-valued logics, with a special attention to three-valued logics. Different approaches to d-valued logics are examined in order to determine some possible universal sets of logic primitives. In particular, we consider the typical connectives of Lukasiewicz and Godel logics, as well as Chang's MV-algebras. As a result, some possible three-valued and d-valued universal gates are described which realize a functionally complete set of fundamental conn…
Quantum logic gates provide fundamental examples of conditional quantum dynamics. They could form the building blocks of general quantum information processing systems which have recently been shown to have many interesting non--classical properties. We describe a simple quantum logic gate, the quantum controlled--NOT, and analyse some of its applications. We discuss two possible physical realisations of the gate; one based on Ramsey atomic interferometry and the other on the selective driving of optical resonances of two subsystems undergoing a dipole--dipole interaction.
We propose a way for realizing a two-qubit controlled phase gate with superconducting quantum interference devices (SQUIDs) coupled to a superconducting resonator. In this proposal, the two lowest levels of each SQUID serve as the logical states and two intermediate levels of each SQUID are used for the gate realization. We show that neither adjustment of SQUID level spacings during the gate operation nor uniformity in SQUID parameters is required by this proposal. In addition, this proposal does not require the adiabatic passage or a second-order detuning and thus the gate is much faster.
We discuss techniques for producing, manipulating and measureing qubits encoded optically as vacuum and single photon states. We show that a universal set of non-deterministic gates can be constructed using linear optics and photon counting. We investigate the efficacy of a test gate given realistic detector efficiencies.
Fault-tolerant quantum computing requires gates which function correctly despite the presence of errors, and are scalable if the error probability-per-gate is below a threshold value. To date, no method has been described for calculating this probability from measurements on a gate. Here we introduce a technique enabling quantitative benchmarking of quantum-logic gates against fault-tolerance thresholds for any architecture. We demonstrate our technique experimentally using a photonic entangling-gate. The relationship between experimental errors and their quantum logic effect is non-trivial: r…
Quantum error correcting codes protect quantum computation from errors caused by decoherence and other noise. Here we study the problem of designing logical operations for quantum error correcting codes. We present an automated procedure which generates logical operations given known encoding and correcting procedures. Our technique is to use variational circuits for learning both the logical gates and the physical operations implementing them. This procedure can be implemented on near-term quantum computers via quantum process tomography. It enables automatic discovery of logical gates from a…
Nonadiabatic holonomic quantum computation has received increasing attention due to its robustness against control errors as well as high-speed realization. The original protocol of nonadiabatic holonomic one-qubit gates has been experimentally demonstrated with superconducting transmon qutrit. However, the original protocol requires two noncommuting gates to realize an arbitrary one-qubit gate, which doubles the exposure time of gates to error sources and therefore makes the gates vulnerable to environment-induced decoherence. Single-shot protocol was subsequently proposed to realize an arbit…
Quantum interrogation can be used as a basic resource for quantum information. This paper presents its applications to entanglement creation and to optical quantum computation. The starting point will be a photon to particle quantum interrogation CZ gate, from which three families of optical CNOT gates will be derived. These gates are nondestructive and can work with probabilities arbitrarily close to 1. A possible experimental implementation with atomic ensembles is also discussed.
We present a simple architecture for deterministic quantum circuits operating on single photon qubits. Few resources are necessary to implement two elementary gates and can be recycled for computing with large numbers of qubits. The deterministic realization of some key multi-qubit gates, such as the Fredkin and Toffoli gate, is greatly simplified in this approach.
The composite rotation approach has been used to develop a range of robust quantum logic gates, including single qubit gates and two qubit gates, which are resistant to systematic errors in their implementation. Single qubit gates based on the BB1 family of composite rotations have been experimentally demonstrated in a variety of systems, but little study has been made of their application in extended computations, and there has been no experimental study of the corresponding robust two qubit gates to date. Here we describe an application of robust gates to Nuclear Magnetic Resonance (NMR) stu…
We propose how to realize a three-step controlled-phase gate of one superconducting qubit simultaneously controlling n qubits selected from N qubits in a cavity (1<n<N). The operation time of this gate is independent of the number n of qubits involved in the gate operation. This phase gate controlling at once n qubits is insensitive to the initial state of the cavity mode and can be used to produce an analogous CNOT gate simultaneously acting on n qubits.
The Toffoli gate is an essential logic element, which permits implementation of a reversible processor. It is of relevance both for classical as well as quantum logics. We propose and theoretically study all-optical implementations of three-bit and four-bit Toffoli gates by application of adiabatic population transfer techniques. For a three-bit Toffoli gate we use variants of stimulated Raman adiabatic passage (STIRAP) processes in a $Λ$-type level scheme, driven by two laser pulses at sufficiently large detunings. For the implementation of a four-bit Toffoli gate, we apply reversible adiabat…
A scheme for coupling superconducting charge qubits via a one-dimensional superconducting transmission line resonator is proposed. The qubits are working at their optimal points, where they are immune to the charge noise and possess long decoherence time. Analysis on the dynamical time evolution of the interaction is presented, which is shown to be insensitive to the initial state of the resonator field. This scheme enables fast gate operation and is readily scalable to multiqubit scenario.
We study two-level q-deformed angular momentum states and us- ing q-deformed harmonic oscillators, we provide a framework for con- structing qubits and quantum gates. We also present the construction of some basic quantum gates including CNOT, SWAP, Toffoli and Fredkin.
The linear optics approach to quantum computing has several potential advantages but the logic operations are probabilistic. Here we review the use of the quantum Zeno effect to suppress the intrinsic failure events in these kinds of devices, which would produce deterministic logic operations without the need for ancilla photons or high-efficiency detectors. The potential advantages of implementing Zeno gates using micro-cavities and electromagnetically-induced transparency are discussed.
We develop a multi-valued logic for quantum computing for use in multi-level quantum systems, and discuss the practical advantages of this approach for scaling up a quantum computer. Generalizing the methods of binary quantum logic, we establish that arbitrary unitary operations on any number of d-level systems (d > 2) can be decomposed into logic gates that operate on only two systems at a time. We show that such multi-valued logic gates are experimentally feasible in the context of the linear ion trap scheme for quantum computing. By using d levels in each ion in this scheme, we reduce the n…
A controlled-NOT logic gate based on conditional spectroscopy has been demonstrated recently for a pair of superconducting flux qubits [Plantenberg et al., Nature 447, 836 (2007)]. Here we study the fidelity of this type of gate applied to a phase qubit coupled to a resonator (or a pair of capacitively coupled phase qubits). Our results show that an intrinsic fidelity of more than 99% is achievable in 45ns.
An implementation is proposed of single qubit gates, e.g., phase, NOT, \sqrt{NOT} and Hadamard, operating on polarized photons and based on light storage. Instead of processing photons themselves, qubit transformations are performed on atomic excitations due to photon storage in a medium of atoms in the tripod configuration.
Logical gates studied in quantum computation suggest a natural logical abstraction that gives rise to a new form of unsharp quantum logic. We study the logical connectives corresponding to the following gates: the Toffoli gate, the NOT and the squareroot of NOT (which admit of natural physical models). This leads to a semantic characterization of a logic that we call computational quantum logic CQL.
It is shown that the two qubit CNOT (controlled NOT) gate can also be realised using q-deformed angular momentum states via the Jordan-Schwinger mechanism.Thus all the three gates necessary for universality i.e. Hadamard, Phase Shift and the two qubit CNOT gate are realisable with q-deformed oscillators.
We introduce and demonstrate experimentally: (1) a framework called "gate set tomography" (GST) for self-consistently characterizing an entire set of quantum logic gates on a black-box quantum device; (2) an explicit closed-form protocol for linear-inversion gate set tomography (LGST), whose reliability is independent of pathologies such as local maxima of the likelihood; and (3) a simple protocol for objectively scoring the accuracy of a tomographic estimate without reference to target gates, based on how well it predicts a set of testing experiments. We use gate set tomography to characteriz…
Fast robust two-qubit gate operation with low susceptibility to crosstalk are the key to scalable quantum information processing. Parametrically driven gate is inherently insensitive to crosstalk while superadiabatic control can speed up the gate without losing accuracy. We propose and experimentally implement superadiabatic two-qubit gates using parametric modulation on superconducting quantum circuits. Our results demonstrate the preservation of adiabaticity at a gate speed close to the quantum limit, in addition to robustness against control instability. We demonstrate a CZ gate with error …
We derive a set of composite pulse sequences that generates CNOT gates and correct all systematic errors within the logical subspace to arbitrary order. These sequences are applicable for any two-qubit interaction Hamiltonian, and make no assumptions about the underlying noise mechanism except that it is constant on the timescale of the operation. We do assume access to error-free single-qubit gates, so single-qubit gate imperfections eventually limit the achievable fidelity. However, since single-qubit gates generally have much higher fidelities than two-qubit gates in practice, these pulse s…
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 demonstrate a robust implementation of a deterministic linear-optical Controlled-NOT (CNOT) gate for single-photon two-qubit quantum logic. A polarization Sagnac interferometer with an embedded 45$^{\circ}$-oriented dove prism is used to enable the polarization control qubit to act on the momentum (spatial) target qubit of the same photon. The CNOT gate requires no active stabilization because the two spatial modes share a common path, and it is used to entangle the polarization and momentum qubits.
High-quality-factor 3D cavities in superconducting circuits are ideal candidates for bosonic logical qubits as their fidelity is limited only by the low photon loss rate. However, the transmon qubits that are used to manipulate bosonic qubits result in the emergence of additional relaxation and dephasing channels. In this work, a numerical study is performed to elucidate the effect of the various loss channels on the performance of logical gates on a bosonic qubit. A gate error model is developed that encapsulates the loss mechanisms for arbitrary gate operations and predicts experimentally ac…
We briefly describe the use of GRAPE pulses to implement quantum logic gates in NMR quantum computers, and discuss a range of simple extensions to the core technique. We then consider a range of difficulties which can arise in practical implementations of GRAPE sequences, reflecting non-idealities in the experimental systems used.
We describe the use of composite rotations to combat systematic errors in single qubit quantum logic gates and discuss three families of composite rotations which can be used to correct off-resonance and pulse length errors. Although developed and described within the context of NMR quantum computing these sequences should be applicable to any implementation of quantum computation.
In this paper we generalize the usual model of quantum computer to a model in which the state is an operator of density matrix and the gates are general superoperators (quantum operations), not necessarily unitary. A mixed state (operator of density matrix) of n two-level quantum system (open or closed n-qubit system) is considered as an element of $4^{n}$-dimensional operator Hilbert space (Liouville space). It allows to use quantum computer (circuit) model with 4-valued logic. The gates of this model are general superoperators which act on n-ququats state. Ququat is quantum state in a 4-dime…
A lower bound on the amount of energy needed to carry out an elementary logical operation on a qubit system, with a given accuracy and in a given time, has been recently postulated. This paper is an attempt to formalize this bound and explore the conditions under which it may be expected to hold. This is a work in progress and any contributions will be appreciated.
Quantum gates based on geometric phases possess intrinsic noise-resilience features and therefore attract much attention. However, the implementations of previous geometric quantum computation typically require a long pulse time of gates. As a result, their experimental control inevitably suffers from the cumulative disturbances of systematic errors due to excessive time consumption. Here, we experimentally implement a set of noncyclic and nonadiabatic geometric quantum gates in a superconducting circuit, which greatly shortens the gate time. And also, we experimentally verify that our univers…
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.
Superconducting qubits are a promising candidate for building a quantum computer. A continued challenge for fast yet accurate gates to minimize the effects of decoherence. Here we apply numerical methods to design fast entangling gates, specifically the controlled Z, in an architecture where two qubits are coupled via a resonator. We find that the gates can be sped up by a factor of two and reach any target fidelity. We also discuss how systematic errors arising from experimental conditions affect the pulses and how to remedy them, providing a strategy for the experimental implementation of ou…
