Renormalization group approach to the P versus NP question

This paper argues that the ideas underlying the renormalization group technique used to characterize phase transitions in condensed matter systems could be useful for distinguishing computational complexity classes. The paper presents a renormalization group transformation that maps an arbitrary Boolean function of $N$ Boolean variables to one of $N-1$ variables. When this transformation is applied repeatedly, the behavior of the resulting sequence of functions is different for a generic Boolean function than for Boolean functions that can be written as a polynomial of degree $\xi$ with $\xi \ll N$ as well as for functions that depend on composite variables such as the arithmetic sum of the inputs. Being able to demonstrate that functions are non-generic is of interest because it suggests an avenue for constructing an algorithm capable of demonstrating that a given Boolean function cannot be computed using resources that are bounded by a polynomial of $N$.Comment: Original version had a conjecture that is known to be false. Revised version corrects this erro

A scalar model of inhomogeneous elastic and granular media

We investigate theoretically how the stress propagation characteristics of granular materials evolve as they are subjected to increasing pressures, comparing the results of a two-dimensional scalar lattice model to those of a molecular dynamics simulation of slightly polydisperse discs. We characterize the statistical properties of the forces using the force histogram and a two-point spatial correlation function of the forces. For the lattice model, in the granular limit the force histogram has an exponential tail at large forces, while in the elastic regime the force histogram is much narrower and has a form that depends on the realization of disorder in the model. The behavior of the force histogram in the molecular dynamics simulations as the pressure is increased is very similar to that displayed by the lattice model. In contrast, the spatial correlations evolve qualitatively differently in the lattice model and in the molecular dynamics simulations. For the lattice model, in the granular limit there are no in-plane stress-stress correlations, whereas in the molecular dynamics simulation significant in-plane correlations persist to the lowest pressures studied.Comment: 28 pages, 29 figures; submitted to PR

Achieving High Fidelity Single Qubit Gates in a Strongly Driven Silicon Quantum Dot Hybrid Qubit

Performing qubit gate operations as quickly as possible can be important to minimize the effects of decoherence. For resonant gates, this requires applying a strong ac drive. However, strong driving can present control challenges by causing leakage to levels that lie outside the qubit subspace. Strong driving can also present theoretical challenges because preferred tools such as the rotating wave approximation can break down, resulting in complex dynamics that are difficult to control. Here we analyze resonant $X$ rotations of a silicon quantum double dot hybrid qubit within a dressed-state formalism, obtaining results beyond the rotating wave approximation. We obtain analytic formulas for the optimum driving frequency and the Rabi frequency, which both are affected by strong driving. While the qubit states exhibit fast oscillations due to counter-rotating terms and leakage, we show that they can be suppressed to the point that gate fidelities above $99.99\%$ are possible, in the absence of decoherence. Hence decoherence mechanisms, rather than strong-driving effects, should represent the limiting factor for resonant-gate fidelities in quantum dot hybrid qubits

Magnetic field dependence of valley splitting in realistic Si/SiGe quantum wells

The authors investigate the magnetic field dependence of the energy splitting between low-lying valley states for electrons in a Si/SiGe quantum well tilted with respect to the crystallographic axis. The presence of atomic steps at the quantum well interface may explain the unexpected, strong suppression of the valley splitting observed in recent experiments. The authors find that the suppression is caused by an interference effect associated with multiple steps, and that the magnetic field dependence arises from the lateral confinement of the electronic wave function. Using numerical simulations, the authors clarify the role of step disorder, obtaining quantitative agreement with the experiments.Comment: Published versio

Exchange in a silicon-based quantum dot quantum computer architecture

In bulk silicon, intervalley electronic interference has been shown to lead to strong oscillations in the exchange coupling between impurity electronic wavefunctions, posing a serious manufacturability problem for proposed quantum computers. Here we show that this problem does not arise in proposed architectures using Si/SiGe quantum dots because of the large in-plane strain in Si quantum wells together with the strong confinement potential typical of heterostructures.Comment: 4 pages, 2 figures, submitted to AP

High-fidelity gates in quantum dot spin qubits

Several logical qubits and quantum gates have been proposed for semiconductor quantum dots controlled by voltages applied to top gates. The different schemes can be difficult to compare meaningfully. Here we develop a theoretical framework to evaluate disparate qubit-gating schemes on an equal footing. We apply the procedure to two types of double-dot qubits: the singlet-triplet (ST) and the semiconducting quantum dot hybrid qubit. We investigate three quantum gates that flip the qubit state: a DC pulsed gate, an AC gate based on logical qubit resonance (LQR), and a gate-like process known as stimulated Raman adiabatic passage (STIRAP). These gates are all mediated by an exchange interaction that is controlled experimentally using the interdot tunnel coupling $g$ and the detuning $\epsilon$, which sets the energy difference between the dots. Our procedure has two steps. First, we optimize the gate fidelity ($f$) for fixed $g$ as a function of the other control parameters; this yields an $f^\text{opt}(g)$ that is universal for different types of gates. Next, we identify physical constraints on the control parameters; this yields an upper bound $f^\text{max}$ that is specific to the qubit-gate combination. We show that similar gate fidelities ($\sim 99.5$%) should be attainable for ST qubits in isotopically purified Si, and for hybrid qubits in natural Si. Considerably lower fidelities are obtained for GaAs devices, due to the fluctuating magnetic fields $\Delta B$ produced by nuclear spins.Comment: 20 pages, 7 figure

Reversible Boolean Networks I: Distribution of Cycle Lengths

We consider a class of models describing the dynamics of $N$ Boolean variables, where the time evolution of each depends on the values of $K$ of the other variables. Previous work has considered models with dissipative dynamics. Here we consider time-reversible models, which necessarily have the property that every possible point in the state-space is an element of one and only one cycle. As in the dissipative case, when K is large, typical orbit lengths grow exponentially with N, whereas for small enough K, typical orbit lengths grow much more slowly with N. The numerical data are consistent with the existence of a phase transition at which the average orbit length grows as a power of N at a value of K between 1.4 and 1.7. However, in the reversible models the interplay between the discrete symmetry and quenched randomness can lead to enormous fluctuations of orbit lengths and other interesting features that are unique to the reversible case. The orbits can be classified by their behavior under time reversal. The orbits that transform into themselves under time reversal have properties quite different from those that do not; in particular, a significant fraction of latter-type orbits have lengths enormously longer than orbits that are time reversal-symmetric. For large K and moderate N, the vast majority of points in the state-space are on one of the time reversal singlet orbits, and a random hopping model gives an accurate description of orbit lengths. However, for any finite K, the random hopping approximation fails qualitatively when N is large enough ($N\gg 2^{(2^K)}$).Comment: 28 pages, 9 figures, Research paper sumbitte

Physically-motivated dynamical algorithms for the graph isomorphism problem

We investigate classical and quantum physics-based algorithms for solving the graph isomorphism problem. Our work integrates and extends previous work by Gudkov et al. (cond-mat/0209112) and by Rudolph (quant-ph/0206068). Gudkov et al. propose an algorithm intended to solve the graph isomorphism problem in polynomial time by mimicking a classical dynamical many-particle process. We show that this algorithm fails to distinguish pairs of non-isomorphic strongly regular graphs, thus providing an infinite class of counterexamples. We also show that the simplest quantum generalization of the algorithm also fails. However, by combining Gudkov et al.'s algorithm with a construction proposed by Rudoph in which one examines a graph describing the dynamics of two particles on the original graph, we find an algorithm that successfully distinguishes all pairs of non-isomorphic strongly regular graphs that we tested (with up to 29 vertices).Comment: 17 pages, 5 figures, 1 table, with additions and corrections include

Analysis of a Population Genetics Model with Mutation, Selection, and Pleiotropy

We investigate the behavior of a population genetics model introduced by Waxman and Peck incorporating mutation, selection, and pleiotropy. The population is infinite and continuous variation of genotype is allowed. Nonetheless, Waxman and Peck showed that if the degree of pleiotropy is large enough,in this model a nonzero fraction of the population can have identical alleles. This `condensed mode' behavior appears in the limit of infinite times. This paper explores the time-dependence of the distribution of alleles in this model. First, the model is analyzed using a recursion technique which enables the distribution of alleles to be calculated at finite times as well as in Waxman and Peck's infinite-time limit. Second, both Waxman and Peck's original model and a related model in which mutations occur continuously are mapped onto problems in quantum mechanics. In both cases, the long-time analysis for the biological model is equivalent to finding the nature of the eigenstates of the quantum problem. The condensed mode appears if and only if there is no bound state in the quantum problem. We compare the behavior of the discrete- and continuous-time versions of the model. The results for the two cases are qualitatively similar, though there are some quantitative differences. We also discuss our attempts to correlate the statistics of DNA sequence variations with the degree of pleiotropy of various genes.Comment: 26 pages, 2 tables, 6 figures, submitted to J. Math. Biolog

A decoherence-free subspace in a charge quadrupole qubit

Quantum computing promises significant speed-up for certain types of computational problems. However, robust implementations of semiconducting qubits must overcome the effects of charge noise that currently limit coherence during gate operations. Here we describe a scheme for protecting solid-state qubits from uniform electric field fluctuations by generalizing the concept of a decoherence-free subspace for spins, and we propose a specific physical implementation: a quadrupole charge qubit formed in a triple quantum dot. The unique design of the quadrupole qubit enables a particularly simple pulse sequence for suppressing the effects of noise during gate operations. Simulations yield gate fidelities 10-1,000 times better than traditional charge qubits, depending on the magnitude of the environmental noise. Our results suggest that any qubit scheme employing Coulomb interactions (for example, encoded spin qubits or two-qubit gates) could benefit from such a quadrupolar design.Comment: 10 pages, 3 figures, Supplementary Information included as appendice