Frank Gaitan

Quantum Error Correction and Fault Tolerant Quantum Computing

It was once widely believed that quantum computation would never become a reality. However, the discovery of quantum error correction and the proof of the accuracy threshold theorem nearly ten years ago gave rise to extensive development and research aimed at creating a working, scalable quantum computer. Over a decade has passed since this monumental accomplishment yet no book-length pedagogical presentation of this important theory exists. Quantum Error Correction and Fault Tolerant Quantum Computing offers the first full-length exposition on the realization of a theory once thought impossible. It provides in-depth coverage on the most important class of codes discovered to datequantum stabilizer codes. It brings together the central themes of quantum error correction and fault-tolerant procedures to prove the accuracy threshold theorem for a particular noise error model. The author also includes a derivation of well-known bounds on the parameters of quantum error correcting code. Packed with over 40 real-world problems, 35 field exercises, and 17 worked-out examples, this book is the essential resource for any researcher interested in entering the quantum field as well as for those who want to understand how the unexpected realization of quantum computing is possible.

Experimental determination of Ramsey numbers.

Ramsey theory is a highly active research area in mathematics that studies the emergence of order in large disordered structures. Ramsey numbers mark the threshold at which order first appears and are extremely difficult to calculate due to their explosive rate of growth. Recently, an algorithm that can be implemented using adiabatic quantum evolution has been proposed that calculates the two-color Ramsey numbers R(m,n). Here we present results of an experimental implementation of this algorithm and show that it correctly determines the Ramsey numbers R(3,3) and R(m,2) for 4≤m≤8. The R(8,2) computation used 84 qubits of which 28 were computational qubits. This computation is the largest experimental implementation of a scientifically meaningful adiabatic evolution algorithm that has been done to date.

Graph isomorphism and adiabatic quantum computing

problem in computer science and is thought to be of comparable difficulty to integer factorization. In this paper we present a quantum algorithm that solves arbitrary instances of GI and which also provides an approach to determiningallautomorphismsofagivengraph.WeshowhowtheGIproblemcanbeconvertedtoacombinatorial optimization problem that can be solved using adiabatic quantum evolution. We numerically simulate the algorithm’s quantum dynamics and show that it correctly (i) distinguishes nonisomorphic graphs; (ii) recognizes isomorphic graphs and determines the permutation(s) that connect them; and (iii) finds the automorphism group of a given graph G. We then discuss the GI quantum algorithm’s experimental implementation, and close by showing how it can be leveraged to give a quantum algorithm that solves arbitrary instances of the NP-complete subgraph isomorphism problem. The computational complexity of an adiabatic quantum algorithm is largely determined by the minimum energy gap � (N) separating the ground and first-excited states in the limit of large problem size N � 1. Calculating � (N) in this limit is a fundamental open problem in adiabatic quantum computing, and so it is not possible to determine the computational complexity of adiabatic quantum algorithms in general, nor consequently, of the specific adiabatic quantum algorithms presented here. Adiabatic quantum computing has been shown to be equivalent to the circuit model of quantum computing, and so development of adiabatic quantum algorithms continues to be of great interest.

Simulation of Quantum Adiabatic Search in the Presence of Noise

Results are presented of a large-scale simulation of the quantum adiabatic search (QuAdS) algorithm in the presence of noise. The algorithm is applied to the NP-Complete problem N-Bit Exact Cover 3 (EC3). The noise is assumed to Zeeman-couple to the qubits and its effects on the algorithms performance is studied for various levels of noise power, and for four different types of noise polarization. We examine the scaling relation between the number of bits N (EC3 problem-size) and the algorithms noise-averaged median run-time 〈T(N)〉. Clear evidence is found of the algorithms sensitivity to noise. Two fits to the simulation results were done: (i) power-law scaling 〈T(N)〉 = aNb; and (ii) exponential scaling 〈T(N)〉 = a[exp(bN) - 1]. Both types of scaling relations provided excellent fits, although the scaling parameters a and b varied with noise power, and with the type of noise polarization. The sensitivity of the scaling exponent b to noise polarization allows a relative assessment of which noise polarizations are most problematic for quantum adiabatic search. We demonstrate how the noise leads to decoherence in QuAdS, and estimate the amount of decoherence present in our simulations. An upper bound is also derived for the noise-averaged QuAdS success probability in the limit of weak noise that is appropriate for our simulations.

Temporal interferometry: A mechanism for controlling qubit transitions during twisted rapid passage with possible application to quantum computing

In an adiabatic rapid passage experiment, the Bloch vector of a two-level system ~qubit! is inverted by slowly inverting an external field to which it is coupled, and along which it is initially aligned. In twisted rapid passage, the external field is allowed to twist around its initial direction with azimuthal angle f(t) at the same time that it is inverted. For polynomial twist, f(t);Bt n . We show that for n>3, multiple avoided crossings can occur during the inversion of the external field, and that these crossings give rise to strong interference effects in the qubit transition probability. The transition probability is found to be a function of the twist strength B, which can be used to control the time separation of the avoided crossings, and hence the character of the interference. Constructive and destructive interference are possible. The interference effects are a consequence of the temporal phase coherence of the wave function. The ability to vary this coherence by varying the temporal separation of the avoided crossings renders twisted rapid passage with adjustable twist strength into a temporal interferometer through which qubit transitions can be greatly enhanced or suppressed. Possible application of this interference mechanism to construction of fast fault-tolerant quantum controlled-NOT and NOT gates is discussed.

Non-adiabatic rapid passage

Experimental verification is presented of recent theoretical predictions of spin inversion schemes that rely on temporal interferences during the sweep. The experiments were carried out using liquid state NMR. It is shown how under non-adiabatic conditions, both complete inversion and complete non-inversion can be achieved, due to slight but easily reproducible alteration of one control parameter. The implications of this scheme for quantum computing are discussed.

Microscopic analysis of the nondissipative force on a line vortex in a superconductor: Berry’s phase, momentum flow, and the Magnus force

A microscopic analysis of the non-dissipative force

Robust high-fidelity universal set of quantum gates through non-adiabatic rapid passage

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Improving quantum gate performance through neighboring optimal control

acting on a line vortex in a type-II superconductor at

Controlling qubit transitions during non-adiabatic rapid passage through quantum interference

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