Pawel Wocjan

Thermodynamic Cost of Reliability and Low Temperatures: Tightening Landauer's Principle and the Second Law

Landauers principle states that the erasure of one bit of information requires thefree energy kT ln 2. We argue that the reliability of the bit erasure process isbounded by the accuracy inherent in the statistical state of the energy source(“the resources”) driving the process. We develop a general framework describingthe “thermodynamic worth” of the resources with respect to reliable bit erasureor good cooling. This worth turns out to be given by the distinguishability of theresources state from its equilibrium state in the sense of a statistical inferenceproblem. Accordingly, Kullback—Leibler relative information is a decisivequantity for the “worth” of the resources state. Due to the asymmetry of relativeinformation, the reliability of the erasure process is bounded rather by the relativeinformation of the equilibrium state with respect to the actual state than by therelative information of the actual state with respect to the equilibrium state (whichis the free energy up to constants).

Sampling from the Thermal Quantum Gibbs State and Evaluating Partition Functions with a Quantum Computer

We present a quantum algorithm to prepare the thermal Gibbs state of interacting quantum systems. This algorithm sets a universal upper bound D(alpha) on the thermalization time of a quantum system, where D is the systems Hilbert space dimension and alpha < or = 1/2 is proportional to the Helmholtz free energy density. We also derive an algorithm to evaluate the partition function of a quantum system in a time proportional to the systems thermalization time and inversely proportional to the targeted accuracy squared.

Hamiltonian quantum cellular automata in one dimension

We construct a simple translationally invariant, nearest-neighbor Hamiltonian on a chain of ten-dimensional qudits that makes it possible to realize universal quantum computing without any external control during the computational process. We only require the ability to prepare an initial computational basis state that encodes both the quantum circuit and its input. The computational process is then carried out by the autonomous Hamiltonian time evolution. After a time polynomially long in the size of the quantum circuit has passed, the result of the computation is obtained with high probability by measuring a few qudits in the computational basis. This result also implies that there cannot exist efficient classical simulation methods for generic translationally invariant nearest-neighbor Hamiltonians on qudit chains, unless quantum computers can be efficiently simulated by classical computers (or, put in complexity theoretic terms, unless

Ergodic Quantum Computing

\mathrm{BPP}=\mathrm{BQP}

Preparing Ground States of Quantum Many-Body Systems on a Quantum Computer

).

Quantum algorithm for approximating partition functions

We propose a (theoretical) model for quantum computation where the result can be read out from the time average of the Hamiltonian dynamics of a 2-dimensional crystal on a cylinder.The Hamiltonian is a spatially local interaction among Wigner–Seitz cells containing six qubits. The quantum circuit that is simulated is specified by the initialization of program qubits. As in Margolus’ Hamiltonian cellular automaton (implementing classical circuits), a propagating wave in a clock register controls asynchronously the application of the gates. However, in our approach all required initializations are basis states. After a while the synchronizing wave is essentially spread around the whole crystal. The circuit is designed such that the result is available with probability about 1/4 despite of the completely undefined computation step. This model reduces quantum computing to preparing basis states for some qubits, waiting, and measuring in the computational basis. Even though it may be unlikely to find our specific Hamiltonian in real solids, it is possible that also more natural interactions allow ergodic quantum computing.

Simulating Hamiltonians in quantum networks: Efficient schemes and complexity bounds

Preparing the ground state of a system of interacting classical particles is an NP-hard problem. Thus, there is in general no better algorithm to solve this problem than exhaustively going through all N configurations of the system to determine the one with lowest energy, requiring a running time proportional to N. A quantum computer, if it could be built, could solve this problem in time sqrt[N]. Here, we present a powerful extension of this result to the case of interacting quantum particles, demonstrating that a quantum computer can prepare the ground state of a quantum system as efficiently as it does for classical systems.

New measures of graph irregularity

We achieve a quantum speed-up of fully polynomial randomized approximation schemes (FPRAS) for estimating partition functions that combine simulated annealing with the Monte-Carlo Markov Chain method and use non-adaptive cooling schedules. The improvement in time complexity is twofold: a quadratic reduction with respect to the spectral gap of the underlying Markov chains and a quadratic reduction with respect to the parameter characterizing the desired accuracy of the estimate output by the FPRAS. Both reductions are intimately related and cannot be achieved separately. First, we use Grover’s fixed point search, quantum walks and phase estimation to efficiently prepare approximate coherent encodings of stationary distributions of the Markov chains. The speed-up we obtain in this way is due to the quadratic relation between the spectral and phase gaps of classical and quantum walks. Second, we generalize the method of quantum counting, showing how to estimate expected values of quantum observables. Using this method instead of classical sampling, we obtain the speed-up with respect to accuracy.

A Simple PromiseBQP-complete Matrix Problem

We address the problem of simulating pair-interaction Hamiltonians in n-node quantum networks where the subsystems have arbitrary, possibly different, dimensions. We show that any pair interaction can be used to simulate any other by applying sequences of appropriate local control sequences. Conditions on time optimal simulation are formulated in terms of spectral majorization of matrices characterizing the coupling parameters. Efficient schemes for decoupling and time reversal can be constructed from orthogonal arrays provided that the dimensions of all nodes are equal to the same prime power. Moreover, we consider a specific system of n harmonic oscillators with bilinear interaction. In this case, decoupling can efficiently be achieved using the combinatorial concept of difference schemes. For this type of interaction we present optimal schemes for inversion.

The 2-local Hamiltonian problem encompasses NP

In this paper, we define and compare four new measures of graph irregularity. We use these measures to prove upper bounds for the chromatic number and the Colin de Verdiere parameter. We also strengthen the concise Turan theorem for irregular graphs and investigate to what extent Turans theorem can be similarly strengthened for generalized r-partite graphs. We conclude by relating these new measures to the Randic index and using the measures to devise new normalised indices of network heterogeneity.