arXiv subjects

Frank Gaitan

Publications and source records attributed to Frank Gaitan.

At least 19 recordsLinked to original sources

arXiv2017

High-fidelity quantum state preparation using neighboring optimal control

We present an approach to single-shot high-fidelity preparation of an $n$-qubit state based on neighboring optimal control theory. This represents a new application of the neighboring optimal control formalism which was originally developed to produce single-shot high-fidelity quantum gates. To illustrate the approach, and to provide a proof-of-principle, we use it to prepare the two qubit Bell state $|β_{01}\rangle = (1/\sqrt{2})\left[\, |01\rangle + |10\rangle\,\right]$ with an error probability $ε\sim 10^{-6}$ ($10^{-5}$) for ideal (non-ideal) control. Using standard methods in the literature, these high-fidelity Bell states can be leveraged to fault-tolerantly prepare the logical state $|\overlineβ_{01}\rangle$.

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arXiv2016

Generalized Ramsey numbers through adiabatic quantum optimization

Ramsey theory is an active research area in combinatorics whose central theme is the emergence of order in large disordered structures, with Ramsey numbers marking the threshold at which this order first appears. For generalized Ramsey numbers $r(G,H)$, the emergent order is characterized by graphs $G$ and $H$. In this paper we: (i) present a quantum algorithm for computing generalized Ramsey numbers by reformulating the computation as a combinatorial optimization problem which is solved using adiabatic quantum optimization; and (ii) determine the Ramsey numbers $r(\mathcal{T}_{m},\mathcal{T}_{n})$ for trees of order $m,n = 6,7,8$, most of which were previously unknown.

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arXiv2014

The Distribution of Ramsey Numbers

We prove that the number of integers in the interval [0,x] that are non-trivial Ramsey numbers r(k,n) (3 <= k <= n) has order of magnitude (x ln x)**(1/2).

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arXiv2014

Improving quantum gate performance through neighboring optimal control

Successful implementation of a fault-tolerant quantum computation on a system of qubits places severe demands on the hardware used to control the many-qubit state. It is known that an accuracy threshold $P_{a}$ exists for any quantum gate that is to be used in such a computation. Specifically, the error probability $P_{e}$ for such a gate must fall below the accuracy threshold: $P_{e} < P_{a}$. Estimates of $P_{a}$ vary widely, though $P_{a}\sim 10^{-4}$ has emerged as a challenging target for hardware designers. In this paper we present a theoretical framework based on neighboring optimal control that takes as input a good quantum gate and returns a new gate with better performance. We illustrate this approach by applying it to all gates in a universal set of quantum gates produced using non-adiabatic rapid passage that has appeared in the literature. Performance improvements are substantial, both for ideal and non-ideal controls. Under suitable conditions detailed below, all gate error probabilities fall well below the target threshold of $10^{-4}$.

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arXiv2014

Graph isomorphism and adiabatic quantum computing

In the Graph Isomorphism problem two N-vertex graphs G and G' are given and the task is to determine whether there exists a permutation of the vertices of G that preserves adjacency and transforms G into G'. If yes, then G and G' are said to be isomorphic; otherwise they are non-isomorphic. The GI problem is an important 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 can also determine all automorphisms of a given graph. We show how the GI problem can be converted to a combinatorial 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 non-isomorphic graphs; (ii) recognizes isomorphic graphs; and (iii) finds all automorphisms 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 Sub-Graph Isomorphism problem.

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arXiv2013

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 $4\leq m\leq 8$. 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.

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arXiv2012

Ramsey numbers and adiabatic quantum computing

The graph-theoretic Ramsey numbers are notoriously difficult to calculate. In fact, for the two-color Ramsey numbers $R(m,n)$ with $m,n\geq 3$, only nine are currently known. We present a quantum algorithm for the computation of the Ramsey numbers $R(m,n)$. We show how the computation of $R(m,n)$ can be mapped to a combinatorial optimization problem whose solution can be found using adiabatic quantum evolution. We numerically simulate this adiabatic quantum algorithm and show that it correctly determines the Ramsey numbers R(3,3) and R(2,s) for $5\leq s\leq 7$. We then discuss the algorithm's experimental implementation, and close by showing that Ramsey number computation belongs to the quantum complexity class QMA.

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arXiv2011

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

We show how a robust high-fidelity universal set of quantum gates can be implemented using a single form of non-adiabatic rapid passage whose parameters are optimized to maximize gate fidelity and reward gate robustness. Each gate in the universal set is found to operate with a fidelity F in the range 0.99988 < F < 0.99999, and to require control parameters with no more than 14-bit (1 part in 10,000) precision. Such precision is within reach of commercially available arbitrary waveform generators, so that an experimental study of this approach to high-fidelity universal quantum control appears feasible.

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arXiv2010

High-fidelity universal quantum gates through group-symmetrized rapid passage

Twisted rapid passage is a type of non-adiabatic rapid passage that generates controllable quantum interference effects that were first observed experimentally in 2003. It is shown that twisted rapid passage sweeps can be used to implement a universal set of quantum gates that operate with high-fidelity. The gate set consists of the Hadamard and NOT gates, together with variants of the phase, pi/8, and controlled-phase gates. For each gate g in the universal set, sweep parameter values are provided which numerical simulations indicate will produce a unitary operation that approximates g with error probability less than 10**(-4). Note that all gates in the universal set are implemented using a single family of control-field, and the error probability for each gate falls below the rough-and-ready estimate for the accuracy threshold of 10**(-4).

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arXiv2009

Density functional theory and quantum computation

This paper establishes the applicability of density functional theory methods to quantum computing systems. We show that ground-state and time-dependent density functional theory can be applied to quantum computing systems by proving the Hohenberg-Kohn and Runge-Gross theorems for a fermionic representation of an N qubit system. As a first demonstration of this approach, time-dependent density functional theory is used to determine the minimum energy gap Delta(N) arising when the quantum adiabatic evolution algorithm is used to solve instances of the NP-Complete problem MAXCUT. It is known that the computational efficiency of this algorithm is largely determined by the large-N scaling behavior of Delta(N), and so determining this behavior is of fundamental significance. As density functional theory has been used to study quantum systems with N ~ 1000 interacting degrees of freedom, the approach introduced in this paper raises the realistic prospect of evaluating the gap Delta(N) for systems with N ~ 1000 qubits. Although the calculation of Delta(N) serves to illustrate how density functional theory methods can be applied to problems in quantum computing, the approach has a much broader range and shows promise as a means for determining the properties of very large quantum computing systems.

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arXiv2009

High-fidelity universal quantum gates through quantum interference

Twisted rapid passage is a type of non-adiabatic rapid passage that gives rise to controllable quantum interference effects that were first observed experimentally in 2003. We show that twisted rapid passage sweeps can be used to implement a universal set of quantum gates that operate with high-fidelity. For each gate in the universal set, sweep parameter values are provided which simulations indicate will yield a quantum gate with error probability P_{e} < 10**(-4). Note that all gates in this universal set are driven by a **single** type of control field (twisted rapid passage), and the error probability for each gate falls below the rough-and-ready estimate for the accuracy threshold P_{a} ~ 10**(-4). The simulations suggest that the universal gate set produced by twisted rapid passage shows promise for use in a fault-tolerant scheme for quantum computing.

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arXiv2008

Noise-Induced Sampling of Alternative Hamiltonian Paths in Quantum Adiabatic Search

We numerically simulate the effects of noise-induced sampling of alternative Hamiltonian paths on the ability of quantum adiabatic search (QuAdS) to solve randomly generated instances of the NP-Complete problem N-bit Exact Cover 3. The noise-averaged median runtime is determined as the noise-power and number of bits N are varied, and power-law and exponential fits are made to the data. Noise is seen to slowdown QuAdS, though a downward shift in the scaling exponent is found for N > 12 over a range of noise-power values. We discuss whether this shift might be connected to arguments in the literature that suggest that altering the Hamiltonian path might benefit QuAdS performance.

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arXiv2007

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 per operation P < 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.

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arXiv2006

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 Exact Cover 3 (EC3). The noise is assumed to Zeeman-couple to the qubits and its effects on the algorithm's performance is studied for various levels of noise power, and for 4 different types of noise polarization. We examine the scaling relation between the number of bits N (EC3 problem size) and the algorithm's noise-averaged median run-time . Clear evidence is found of the algorithm's sensitivity to noise. Two fits to the simulation results were done: (1) power-law scaling = aN**b; and (2) exponential scaling = a[exp(bN) - 1]. Both types of scaling relations provided excellent fits. We demonstrate how noise leads to decoherence in QuAdS, estimate the amount of decoherence in our simulations, and derive an upper bound for the noise-averaged QuAdS success probability in the weak noise limit appropriate for our simulations.

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arXiv2004

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

In adiabatic rapid passage, the Bloch vector of a qubit is inverted by slowly inverting an external field to which it is coupled, and along which it is initially aligned. In non-adiabatic twisted rapid passage, the external field is allowed to twist around its initial direction with azimuthal angle ϕ(t) at the same time that it is non-adiabatically inverted. For polynomial twist, ϕ(t) \sim Bt^{n}. We show that for n \ge 3, multiple qubit resonances can occur during a single inversion of the external field, producing strong interference effects in the qubit transition probability. The character of the interference is controllable through variation of the twist strength B. Constructive and destructive interference are possible, greatly enhancing or suppressing qubit transitions. Experimental confirmation of these controllable interference effects has already occurred. Application of this interference mechanism to the construction of fast fault-tolerant quantum CNOT and NOT gates is discussed.

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arXiv2003

Noisy Control, the Adiabatic Geometric Phase, and Destruction of the Efficiency of Geometric Quantum Computation

We examine the adiabatic dynamics of a quantum system coupled to a noisy classical control field. A stochastic phase shift is shown to arise in the off-diagonal elements of the system's density matrix which can cause decoherence. We derive the condition for onset of decoherence, and identify the noise properties that drive decoherence. We show how this decoherence mechansim causes: (1) a dephasing of the observable consequences of the adiabatic geometric phase; and (2) the loss of computational efficiency of the Shor algorithm when run on a sufficiently noisy geometric quantum computer.

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arXiv2003

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 $ϕ(t)$ at the same time that it is inverted. For polynomial twist: $ϕ(t) \sim Bt^{n}$. We show that for $n \geq 3$, 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 wavefunction. 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 CNOT and NOT gates is discussed.

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arXiv1998

Berry's Phase in the Presence of a Non-Adiabatic Environment

We consider a two-level system coupled to an environment that evolves non-adiabatically. We present a non-perturbative method for determining the persistence amplitude whose phase contains all the corrections to Berry's phase produced by the non-adiabatic motion of the environment. Specifically, it includes the effects of transitions between the two energy levels to all orders in the non-adiabatic coupling. The problem of determining all non-adiabatic corrections is reduced to solving an ordinary differential equation to which numerical methods should provide solutions in a variety of situations. We apply our method to a particular example that can be realized as a magnetic resonance experiment, thus raising the possibility of testing our results in the lab.

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