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Daniel A. Lidar

Publications and source records attributed to Daniel A. Lidar.

At least 127 records · Page 7Linked to original sources
arXiv2010

Rigorous Bounds for Optimal Dynamical Decoupling

We present rigorous performance bounds for the optimal dynamical decoupling pulse sequence protecting a quantum bit (qubit) against pure dephasing. Our bounds apply under the assumption of instantaneous pulses and of bounded perturbing environment and qubit-environment Hamiltonians. We show that if the total sequence time is fixed the optimal sequence can be used to make the distance between the protected and unperturbed qubit states arbitrarily small in the number of applied pulses. If, on the other hand, the minimum pulse interval is fixed and the total sequence time is allowed to scale with the number of pulses, then longer sequences need not always be advantageous. The rigorous bound may serve as testbed for approximate treatments of optimal decoupling in bounded models of decoherence.

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arXiv2010

Entanglement and area law with a fractal boundary in a topologically ordered phase

Quantum systems with short range interactions are known to respect an area law for the entanglement entropy: the von Neumann entropy $S$ associated to a bipartition scales with the boundary $p$ between the two parts. Here we study the case in which the boundary is a fractal. We consider the topologically ordered phase of the toric code with a magnetic field. When the field vanishes it is possible to analytically compute the entanglement entropy for both regular and fractal bipartitions $(A,B)$ of the system, and this yields an upper bound for the entire topological phase. When the $A$-$B$ boundary is regular we have $S/p =1$ for large $p$. When the boundary is a fractal of Hausdorff dimension $D$, we show that the entanglement between the two parts scales as $S/p=γ\leq1/D$, and $γ$ depends on the fractal considered.

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arXiv2009

Near-optimal dynamical decoupling of a qubit

We present a near-optimal quantum dynamical decoupling scheme that eliminates general decoherence of a qubit to order n using O(n^2) pulses, an exponential decrease in pulses over all previous decoupling methods. Numerical simulations of a qubit coupled to a spin bath demonstrate the superior performance of the new pulse sequences.

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arXiv2009

Scheme for fault-tolerant holonomic computation on stabilizer codes

This paper generalizes and expands upon the work [Phys. Rev. Lett. 102, 070502 (2009)] where we introduced a scheme for fault-tolerant holonomic quantum computation (HQC) on stabilizer codes. HQC is an all-geometric strategy based on non-Abelian adiabatic holonomies, which is known to be robust against various types of errors in the control parameters. The scheme we present shows that HQC is a scalable method of computation, and opens the possibility for combining the benefits of error correction with the inherent resilience of the holonomic approach. We show that with the Bacon-Shor code the scheme can be implemented using Hamiltonian operators of weight 2 and 3.

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arXiv2009

Fault-tolerant holonomic quantum computation

We explain how to combine holonomic quantum computation (HQC) with fault tolerant quantum error correction. This establishes the scalability of HQC, putting it on equal footing with other models of computation, while retaining the inherent robustness the method derives from its geometric nature.

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arXiv2008

Channel-Optimized Quantum Error Correction

We develop a theory for finding quantum error correction (QEC) procedures which are optimized for given noise channels. Our theory accounts for uncertainties in the noise channel, against which our QEC procedures are robust. We demonstrate via numerical examples that our optimized QEC procedures always achieve a higher channel fidelity than the standard error correction method, which is agnostic about the specifics of the channel. This demonstrates the importance of channel characterization before QEC procedures are applied. Our main novel finding is that in the setting of a known noise channel the recovery ancillas are redundant for optimized quantum error correction. We show this using a general rank minimization heuristic and supporting numerical calculations. Therefore, one can further improve the fidelity by utilizing all the available ancillas in the encoding block.

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arXiv2008

Operator quantum error correction for continuous dynamics

We study the conditions under which a subsystem code is correctable in the presence of noise that results from continuous dynamics. We consider the case of Markovian dynamics as well as the general case of Hamiltonian dynamics of the system and the environment, and derive necessary and sufficient conditions on the Lindbladian and system-environment Hamiltonian, respectively. For the case when the encoded information is correctable during an entire time interval, the conditions we obtain can be thought of as generalizations of the previously derived conditions for decoherence-free subsystems to the case where the subsystem is time dependent. As a special case, we consider conditions for unitary correctability. In the case of Hamiltonian evolution, the conditions for unitary correctability concern only the effect of the Hamiltonian on the system, whereas the conditions for general correctability concern the entire system-environment Hamiltonian. We also derive conditions on the Hamiltonian which depend on the initial state of the environment, as well as conditions for correctability at only a particular moment of time. We discuss possible implications of our results for approximate quantum error correction.

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arXiv2008

Encoding One Logical Qubit Into Six Physical Qubits

We discuss two methods to encode one qubit into six physical qubits. Each of our two examples corrects an arbitrary single-qubit error. Our first example is a degenerate six-qubit quantum error-correcting code. We explicitly provide the stabilizer generators, encoding circuit, codewords, logical Pauli operators, and logical CNOT operator for this code. We also show how to convert this code into a non-trivial subsystem code that saturates the subsystem Singleton bound. We then prove that a six-qubit code without entanglement assistance cannot simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an arbitrary single-qubit error. A corollary of this result is that the Steane seven-qubit code is the smallest single-error correcting CSS code. Our second example is the construction of a non-degenerate six-qubit CSS entanglement-assisted code. This code uses one bit of entanglement (an ebit) shared between the sender and the receiver and corrects an arbitrary single-qubit error. The code we obtain is globally equivalent to the Steane seven-qubit code and thus corrects an arbitrary error on the receiver's half of the ebit as well. We prove that this code is the smallest code with a CSS structure that uses only one ebit and corrects an arbitrary single-qubit error on the sender's side. We discuss the advantages and disadvantages for each of the two codes.

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arXiv2008

Bang-Bang control of a qubit coupled to a quantum critical spin bath

We analytically and numerically study the effects of pulsed control on the decoherence of a qubit coupled to a quantum spin bath. When the environment is critical, decoherence is faster and we show that the control is relatively more effective. Two coupling models are investigated, namely a qubit coupled to a bath via a single link and a spin star model, yielding results that are similar and consistent.

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arXiv2008

Towards Fault Tolerant Adiabatic Quantum Computation

I show how to protect adiabatic quantum computation (AQC) against decoherence and certain control errors, using a hybrid methodology involving dynamical decoupling, subsystem and stabilizer codes, and energy gaps. Corresponding error bounds are derived. As an example I show how to perform decoherence-protected AQC against local noise using at most two-body interactions.

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arXiv2008

Optimal Dynamical Decoherence Control of a Qubit

A theory of dynamical control by modulation for optimal decoherence reduction is developed. It is based on the non-Markovian Euler-Lagrange equation for the energy-constrained field that minimizes the average dephasing rate of a qubit for any given dephasing spectrum.

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arXiv2007

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.

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arXiv2007

Adiabatic Preparation of Topological Order

Topological order characterizes those phases of matter that defy a description in terms of symmetry and cannot be distinguished in terms local order parameters. This type of order plays a key role in the theory of the fractional quantum Hall effect, as well as in topological quantum information processing. Here we show that a system of n spins forming a lattice on a Riemann surface can undergo a second order quantum phase transition between a spin-polarized phase and a string-net condensed phase. This is an example of a phase transition between magnetic and topological order. We furthermore show how to prepare the topologically ordered phase through adiabatic evolution in a time that is upper bounded by O(\sqrt{n}). This provides a physically plausible method for constructing a topological quantum memory. We discuss applications to topological and adiabatic quantum computing.

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arXiv2007

The Spin Density Matrix II: Application to a system of two quantum dots

This work is a sequel to our work "The Spin Density Matrix I: General Theory and Exact Master Equations" (eprint arXiv:0708.0644 [cond-mat]). Here we compare pure- and pseudo-spin dynamics using as an example a system of two quantum dots, a pair of localized conduction-band electrons in an n-doped GaAs semiconductor. Pure-spin dynamics is obtained by tracing out the orbital degrees of freedom, whereas pseudo-spin dynamics retains (as is conventional) an implicit coordinate dependence. We show that magnetic field inhomogeneity and spin-orbit interaction result in a non-unitary evolution in pure-spin dynamics, whereas these interactions contribute to the effective pseudo-spin Hamiltonian via terms that are asymmetric in spin permutations, in particular, the Dzyaloshinskii-Moriya (DM) spin-orbit interaction. We numerically investigate the non-unitary effects in the dynamics of the triplet states population, purity, and Lamb energy shift, as a function of interdot distance and magnetic field difference. The spin-orbit interaction is found to produce effects of roughly four orders of magnitude smaller than those due to magnetic field difference in the pure-spin model. We estimate the spin-orbit interaction magnitude in the DM-interaction term. Our estimate gives a smaller value than that recently obtained by Kavokin [Phys. Rev. B 64, 075305 (2001)], who did not include double occupancy effects. We show that a necessary and sufficient condition for obtaining a universal set of quantum logic gates, involving only two spins, in both pure- and pseudo-spin models is that the magnetic field inhomogeneity and the Heisenberg interaction are both non-vanishing. We also briefly analyze pure-spin dynamics in the electron on liquid helium system recently proposed by Lyon [Phys. Rev. A 74, 052338 (2006)].

cond-mat.str-el↗
arXiv2007

The Spin Density Matrix I: General Theory and Exact Master Equations

We consider a scenario where interacting electrons confined in quantum dots (QDs) are either too close to be resolved, or we do not wish to apply measurements that resolve them. Then the physical observable is an electron spin only (one cannot unambiguously ascribe a spin to a QD) and the system state is fully described by the spin-density matrix. Accounting for the spatial degrees of freedom, we examine to what extent a Hamiltonian description of the spin-only degrees of freedom is valid. We show that as long as there is no coupling between singlet and triplet states this is indeed the case, but when there is such a coupling there are open systems effects, i.e., the dynamics is non-unitary even without interaction with a true bath. Our primary focus is an investigation of non-unitary effects, based on exact master equations we derive for the spin-density matrix in the Lindblad and time-convolutionless (TCL) forms, and the implications for quantum computation. In particular, we demonstrate that the Heisenberg interaction does not affect the unitary part (apart from a Lamb shift) but does affect the non-unitary contributions to time evolution of the spin-density matrix. In a sequel paper we present a detailed analysis of an example system of two quantum dots, including spin-orbit effects.

cond-mat.str-el↗
arXiv2007

Non-Markovian dynamics of a qubit coupled to an Ising spin bath

We study the analytically solvable Ising model of a single qubit system coupled to a spin bath. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the system qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs particularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed post-Markovian master equation performs comparably to the time-convolutionless master equation for a properly chosen memory kernel, and outperforms all the approximation methods considered here at long times. Our findings shed light on the applicability of master equations to the description of reduced system dynamics in the presence of spin-baths.

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