One-qubit Gates Research Articles

Resonant quantum gates in circuit quantum electrodynamics

We propose the implementation of fast resonant gates in circuit quantum electrodynamics for quantum information processing. We show how a suitable utilization of three-level superconducting qubits inside a resonator constitutes a key tool to perform diverse two-qubit resonant gates, improving the operation speed when compared to slower dispersive techniques. To illustrate the benefit of resonant two-qubit gates in circuit QED, we consider the implementation of a two-dimensional cluster state in an array of N x N superconducting qubits by using resonant controlled-phase (CPHASE) and one-qubit gates, where the generation time grows linearly with N. For N=3, and taking into account decoherence mechanisms, a fidelity over 60% for the generation of this cluster state is obtained.

Upper bounds on the noise threshold for fault-tolerant quantum computing

We prove new upper bounds on the tolerable level of noise in a quantum circuit. Weconsider circuits consisting of unitary k-qubit gates each of whose input wires is subject todepolarizing noise of strength p, as well as arbitrary one-qubit gates that are essentiallynoise-free. We assume that the output of the circuit is the result of measuring somedesignated qubit in the final state. Our main result is that for p > 1 - Θ(1/√k), theoutput of any such circuit of large enough depth is essentially independent of its input,thereby making the circuit useless. For the important special case of k = 2, our bound isp > 35.7%. Moreover, if the only allowed gate on more than one qubit is the two-qubitCNOT gate, then our bound becomes 29.3%. These bounds on p are numerically betterthan previous bounds, yet are incomparable because of the somewhat different circuitmodel that we are using. Our main technique is the use of a Pauli basis decomposition,in which the effects of depolarizing noise are very easy to describe.

Upper bounds on the noise threshold for fault-tolerant quantum computing

We prove new upper bounds on the tolerable level of noise in a quantum circuit. We consider circuits consisting of unitary $k$-qubit gates each of whose input wires is subject to depolarizing noise of strength $p$, as well as arbitrary one-qubit gates that are essentially noise-free. We assume that the output of the circuit is the result of measuring some designated qubit in the final state. Our main result is that for $p>1-\Theta(1/\sqrt{k})$, the output of any such circuit of large enough depth is essentially independent of its input, thereby making the circuit useless. For the important special case of $k=2$, our bound is $p>35.7\%$. Moreover, if the only allowed gate on more than one qubit is the two-qubit CNOT gate, then our bound becomes $29.3\%$. These bounds on $p$ are numerically better than previous bounds, yet are incomparable because of the somewhat different circuit model that we are using. Our main technique is the use of a Pauli basis decomposition, in which the effects of depolarizing noise are very easy to describe.

One-way quantum computation with circuit quantum electrodynamics

In this Brief Report, we propose a potential scheme to implement one-way quantum computation with circuit quantum electrodynamics (QED). Large cluster states of charge qubits can be generated in just one step with a superconducting transmission line resonator (TLR) playing the role of a dispersive coupler. A single-qubit measurement in the arbitrary basis can be implemented using a single electron transistor with the help of one-qubit gates. By examining the main decoherence sources, we show that circuit QED is a promising architecture for one-way quantum computation.

Efficient Atomic One-Qubit Phase Gate Realized by a Cavity QED and Identical Atoms System

We present a scheme to implement a one-qubit phase gate with a two-level atom crossing an optical cavity in which some identical atoms are trapped. One can conveniently acquire an arbitrary phase shift of the gate by properly choosing the number of atoms trapped in the cavity and the velocity of the atom crossing the cavity. The present scheme provides a very simple and efficient way for implementing one-qubit phase gate.

Geometric quantum gates in liquid-state NMR based on a cancellation of dynamical phases

A proposal for applying nonadiabatic geometric phases to quantum computing, called double-loop method [S.-L. Zhu and Z. D. Wang, Phys. Rev. A 67, 022319 (2003)], is demonstrated in a liquid-state nuclear magnetic-resonance quantum computer. Using a spin-echo technique, the original method is modified so that quantum gates are implemented in a standard high-precision nuclear magnetic-resonance system for chemical analysis. We show that a dynamical phase is successfully eliminated and a one-qubit quantum gate is realized although the gate fidelity is not high.

Robust and fast geometric quantum computation with multiqubit gates in cavity QED

A scheme is proposed for realizing geometric quantum computation with one-qubit unitary gates and $N$-qubit $(N\ensuremath{\ge}2)$ phase gates in cavity QED. In the scheme, the cavity field and excited states are virtually excited by controlling appropriately atom-field couplings and engineering adiabatic passage; thus the gate operations are insensitive to the cavity decay and the atomic spontaneous emission. Furthermore, the $N$-qubit $(N\ensuremath{\ge}2)$ geometric phase gates can be realized just by one step and the required time does not increase with the number of qubits, which can significantly reduce complexity of physical realization of practical quantum computation and quantum algorithms.

On the CNOT -cost of TOFFOLI gates

The three-input \TOFFOLI\ gate is the workhorse of circuit synthesis for classical logic operations on quantum data, e.g., reversible arithmetic circuits. In physical implementations, however, \TOFFOLI\ gates are decomposed into six \CNOT\ gates and several one-qubit gates. Though this decomposition has been known for at least 10 years, we provide here the first demonstration of its \CNOT-optimality. We study three-qubit circuits which contain less than six \CNOT\ gates and implement a block-diagonal operator, then show that they implicitly describe the cosine-sine decomposition of a related operator. Leveraging the canonical nature of such decompositions to limit one-qubit gates appearing in respective circuits, we prove that the $n$-qubit analogue of the \TOFFOLI\ requires at least $2n$ \CNOT\ gates. Additionally, our results offer a complete classification of three-qubit diagonal operators by their \CNOT -cost, which holds even if ancilla qubits are available.

High fidelity universal set of quantum gates using non-adiabatic rapid passage

Numerical simulation results are presented which suggest that a class of non-adiabatic rapid passage sweeps first realized experimentally in 1991 should be capable of implementing a universal set of quantum gates \uniset\ that operate with high fidelity. The gates constituting \uniset\ are the Hadamard and NOT gates, together with variants of the phase, $\pi /8$, and controlled-phase gates. The universality of \uniset\ is established by showing that it can construct the universal set consisting of Hadamard, phase, $\pi /8$, and controlled-NOT gates. Sweep parameter values are provided which simulations indicate will produce the different gates in \uniset , and for which the gate error probability $P_{e}$ satisfies: (i)~$P_{e}<10^{-4}$ for the one-qubit gates; and (ii)~$P_{e}<1.27\times 10^{-3}$ for the modified controlled-phase gate. The sweeps in this class are non-composite and generate controllable quantum interference effects that allow the gates in \uniset\ to operate non-adiabatically while maintaining high fidelity. These interference effects have been observed using NMR, and it has previously been shown how these rapid passage sweeps can be applied to atomic systems using electric fields. Here we show how these sweeps can be applied to both superconducting charge and flux qubit systems. The simulations suggest that the universal set of gates \uniset\ produced by these rapid passage sweeps shows promise as possible elements of a fault-tolerant scheme for quantum computing.

Spintronic single-qubit gate based on aΠ-shaped lateral quantum dot with spin-orbit interaction

In a quantum nanometric lateral quantum dot (QD) with a $\ensuremath{\Pi}$ geometry the spin properties of a single electron injected in the Coulomb blockade regime are modified by the spin-orbit interaction, resulting in a transformation of the qubit state carried by the spin. The $\ensuremath{\Pi}$-shaped QD acts as a one-qubit spintronic quantum gate whose properties can be varied by tuning the Rashba parameter of the spin-orbit interaction, by changing the relative position of the junctions, as well as by modifying the length of the transverse legs. We discuss how a large class of transformations can be attained with already one $\ensuremath{\Pi}$ nanojunction or a few $\ensuremath{\Pi}$ nanojunctions in series.

Crossover from the ballistic to the resonant tunneling transport for an ideal one-dimensional quantum ring with spin-orbit interaction

Quantum interference effects in nanometric devices provide suitable means for controlling spin at mesoscopic scales. As discussed by Foldi et al. [Phys. Rev. B 71, 033309 (2005)], in a quantum ring connected with two external leads, the spin properties of an incoming electron are modified by the spin-orbit interaction (SOI), resulting in a transformation of the qubit state carried by the spin. The ring acts as a one-qubit spintronic quantum gate whose properties can be varied by tuning the Rashba parameter of the SOI, as well as by changing the size of the ring. However continuous transport experiment cannot be utilized as quantum information processing because in spintronics one has to handle a single electron which carries the information in its spin. Thus, starting from the ballistic structure by Foldi et al., we propose a device which works in the regime of resonant tunneling and is able to handle a single electron. We discuss the crossover between the two different regimes.

Modified Khaneja–Glaser Decomposition and Realization of Three-Qubit Quantum Gate

Optimal implementation of quantum gates is crucial for realization of quantum computation. We slightly modify the Khaneja-Glaser decomposition (KGD) for n-qubits and give a new Cartan subalgbra in the second step of the decomposition. Based on this modified KGD, we investigate the realization of three-qubit logic gate and obtain the result that a general three-qubit quantum logic gate can be implemented using at most 73 one-qubit gates rotations with respect to the y and z axes and 26 CNOT gates.

ANALYTIC ONE-BIT AND CNOT GATE CONSTRUCTIONS OF GENERAL n-QUBIT CONTROLLED GATES

General n-qubit controlled unitary gates are frequently used in quantum information processing tasks. Barenco, Bennett, Cleve, Di Vincenzo, Margolus and Shor [Phys. Rev. A52 (1995) 3457] have given the general construction methods, and explicit results for up-to-four-qubits controlled unitary gates. We extended their calculation and gave two analytic expressions for the construction of general n-qubit controlled unitary gates in terms of one-qubit and two-qubit CNOT gates. There are two expressions – one is exponential in the qubit number which is efficient for up to ten qubits, and the other is polynomial in the qubit number, which is efficient for more than ten qubits.

Efficient one- and two-qubit pulsed gates for an oscillator-stabilized Josephson qubit

We present theoretical schemes for performing high-fidelity one- and two-qubit pulsed gates for a superconducting flux qubit. The ‘IBM qubit’ consists of three Josephson junctions, three loops and a superconducting transmission line. Assuming a fixed inductive qubit–qubit coupling, we show that the effective qubit–qubit interaction is tunable by changing the applied fluxes, and can be made negligible, allowing one to perform high-fidelity single qubit gates. Our schemes are tailored to alleviate errors due to 1/f noise; we find gates with only 1% loss of fidelity due to this source, for pulse times in the range of 20–30 ns for one-qubit gates (Z rotations, Hadamard) and 60 ns for a two-qubit gate (controlled-Z). Our relaxation and dephasing time estimates indicate a comparable loss of fidelity from this source. The control of leakage plays an important role in the design of our shaped pulses, preventing shorter pulse times. However, we have found that imprecision in the control of the quantum phase plays a major role in the limitation of the fidelity of our gates.

Randomized benchmarking of quantum gates

A key requirement for scalable quantum computing is that elementary quantum gates can be implemented with sufficiently low error. One method for determining the error behavior of a gate implementation is to perform process tomography. However, standard process tomography is limited by errors in state preparation, measurement and one-qubit gates. It suffers from inefficient scaling with number of qubits and does not detect adverse error-compounding when gates are composed in long sequences. An additional problem is due to the fact that desirable error probabilities for scalable quantum computing are of the order of 0.0001 or lower. Experimentally proving such low errors is challenging. We describe a randomized benchmarking method that yields estimates of the computationally relevant errors without relying on accurate state preparation and measurement. Since it involves long sequences of randomly chosen gates, it also verifies that error behavior is stable when used in long computations. We implemented randomized benchmarking on trapped atomic ion qubits, establishing a one-qubit error probability per randomized $\ensuremath{\pi}/2$ pulse of 0.00482(17) in a particular experiment. We expect this error probability to be readily improved with straightforward technical modifications.

Fidelity of optimally controlled quantum gates with randomly coupled multiparticle environments

This work studies the feasibility of optimal control of high-fidelity quantum gates in a model of interacting two-level particles. One particle (the qubit) serves as the quantum information processor, whose evolution is controlled by a time-dependent external field. The other particles are not directly controlled and serve as an effective environment, coupling to which is the source of decoherence. The control objective is to generate target one-qubit gates in the presence of strong environmentally-induced decoherence and under physically motivated restrictions on the control field. It is found that interactions among the environmental particles have a negligible effect on the gate fidelity and require no additional adjustment of the control field. Another interesting result is that optimally controlled quantum gates are remarkably robust to random variations in qubit-environment and inter-environment coupling strengths. These findings demonstrate the utility of optimal control for management of quantum-information systems in a very precise and specific manner, especially when the dynamics complexity is exacerbated by inherently uncertain environmental coupling.

High-fidelity single-qubit gates using non-adiabatic rapid passage

Numerical simulation results are presented which suggest that a class of non-adiabatic rapid passage sweeps first realized experimentally in 1991 should be capable of implementing a set of quantum gates that is universal for one-qubit unitary operations and whose elements operate with error probabilities $P_{e}<10^{-4}$. The sweeps are non-composite and generate controllable quantum interference effects which allow the one-qubit gates produced to operate non-adiabatically while maintaining high accuracy. The simulations suggest that the one-qubit gates produced by these sweeps show promise as possible elements of a fault-tolerant scheme for quantum computing.

Geometric phase gates based on stimulated Raman adiabatic passage in tripod systems

We consider stimulated Raman adiabatic passage (STIRAP) processes in tripod systems and show how to generate purely geometric phase changes of the quantum states involved. The geometric phases are controlled by three laser fields where pulse shapes, relative field strength and phases can be controlled. We present a robust set of universal gates for quantum computing based on these geometric phases: a one-qubit phase gate, a Hadamard gate and a two-qubit phase gate.

Precision of single-qubit gates based on Raman transitions

We analyze the achievable precision for single-qubit gates that are based on Raman transitions between two near-degenerate ground states via a virtually excited state. In particular, we study the errors due to non-perfect adiabaticity and due to spontaneous emission from the excited state. For the case of non-adabaticity, we calculate the error as a function of the dimensionless parameter $\chi=\Delta \tau$, where $\Delta$ is the detuning of the Raman beams and $\tau$ is the gate time. For the case of spontaneous emission, we give an analytical argument that the gate errors are approximately equal to $\Lambda \gamma/\Delta$, where $\Lambda$ is the rotation angle of the one-qubit gate and $\gamma$ is the spontaneous decay rate, and we show numerically that this estimate holds to good approximation.

Cluster-state quantum computing in optical fibers

A scheme for the implementation of the cluster-state model of quantum computing in optical fibers, which enables the feed-forward feature, is proposed. This scheme uses the time-bin encoding of qubits. Following previously suggested methods of applying arbitrary one-qubit gates in optical fibers, two different ways for the realization of fusion gate types I and II for cluster production are proposed: a fully time-bin-based encoding scheme and a combination of time-bin and polarization-based encoding. Also, the methods of measurement in any desired bases for the purpose of processing of cluster-state computing for both these encodings are explained.