We compute the expected randomized benchmarking sequence fidelity for a system subject to Gaussian time-correlated noise. For single qubit benchmarking we show that the expected sequence fidelity is given by the partition function of a long-range coupled spin-one Ising model, with each site in the Ising model corresponding to a free evolution interval. For d-state systems, the expected sequence fidelity is given by an Ising-like model partition function whose site variables are given by the weights of the adjoint representation of SU(d). A high effective temperature expansion for the partition function in the single qubit case shows decay of sequence fidelity varying from exponential for uncorrelated noise to a power law for quasistatic noise. Fitting an exponential to the sequence fidelity decay under correlated noise gives unreliable estimates of the average gate error rate.We compute the expected randomized benchmarking sequence fidelity for a system subject to Gaussian time-correlated noise. For single qubit benchmarking we show that the expected sequence fidelity is given by the partition function of a long-range coupled spin-one Ising model, with each site in the Ising model corresponding to a free evolution interval. For d-state systems, the expected sequence fidelity is given by an Ising-like model partition function whose site variables are given by the weights of the adjoint representation of SU(d). A high effective temperature expansion for the partition function in the single qubit case shows decay of sequence fidelity varying from exponential for uncorrelated noise to a power law for quasistatic noise. Fitting an exponential to the sequence fidelity decay under correlated noise gives unreliable estimates of the average gate error rate.

arXiv:1703.09747Mar 28, 20178 citations

GENERALIZED BROWNIAN FUNCTIONALS AND THE FEYNMAN INTEGRAL

Ludwig Streit, T. HIDA

WORLD SCIENTIFIC eBooksApr 1, 20013 citations

Critical properties of many-component systems

Victor J. Emery

Critical properties are discussed for systems with order parameters given by $n$ vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{\ensuremath{\alpha}}$, each with $m$ components. The Hamiltonian has an arbitrary symmetry for each vector separately, but there is a particular kind of coupling between them. It is shown that there is an integral representation for the partition function which reduces $n$ to an explicit parameter in an averaged partition function for the $m$ -component model. This leads to a simple discussion of properties of the system as a function of $n$ . In particular, it is possible to give a coherent derivation of several known and new results without the aid of perturbation theory or the renormalization-group method. It is shown that, in certain special cases, the exponents are Gaussian when $n$ is a negative even integer and that $n=0$ corresponds to the excluded-volume problem. The general case is shown to reduce to an arbitrary $m$ -component model which is random when $n=0$ and constrained when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. A direct derivation of the large- $n$ limit is given and leads to a variety of exactly solvable models. Expressions for the order ${n}^{\ensuremath{-}1}$ correction are obtained in terms of correlation functions. This expansion is valid at all temperatures and for any order of transition, so that it is particularly suitable for considering tricritical phenomena.Critical properties are discussed for systems with order parameters given by $n$ vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{\ensuremath{\alpha}}$, each with $m$ components. The Hamiltonian has an arbitrary symmetry for each vector separately, but there is a particular kind of coupling between them. It is shown that there is an integral representation for the partition function which reduces $n$ to an explicit parameter in an averaged partition function for the $m$-component model. This leads to a simple discussion of properties of the system as a function of $n$. In particular, it is possible to give a coherent derivation of several known and new results without the aid of perturbation theory or the renormalization-group method. It is shown that, in certain special cases, the exponents are Gaussian when $n$ is a negative even integer and that $n=0$ corresponds to the excluded-volume problem. The general case is shown to reduce to an arbitrary $m$-component model which is random when $n=0$ and constrained when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. A direct derivation of the large-$n$ limit is given and leads to a variety of exactly solvable models. Expressions for the order ${n}^{\ensuremath{-}1}$ correction are obtained in terms of correlation functions. This expansion is valid at all temperatures and for any order of transition, so that it is particularly suitable for considering tricritical phenomena.

Physical review. B, Solid stateJan 1, 1975344 citations

Computing partition functions

Paul M. Carter, Gyan Bhanut

Nuclear Physics B - Proceedings SupplementsSep 1, 19881 citation

On the representations of the partition function of a system of interacting bosons as integrals over Gaussian random fields

F.W. Wiegel, J. Hijmans

Jan 1, 19740 citations

The Gaussian Equivalent Representation of Functional Integrals

Lecture notes in physics. M, MonographsDec 15, 20080 citations

Critical properties are discussed for systems with order parameters given by

n

vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{\ensuremath{\alpha}}$, each with

m

components. The Hamiltonian has an arbitrary symmetry for each vector separately, but there is a particular kind of coupling between them. It is shown that there is an integral representation for the partition function which reduces

n

to an explicit parameter in an averaged partition function for the

m

-component model. This leads to a simple discussion of properties of the system as a function of

n

. In particular, it is possible to give a coherent derivation of several known and new results without the aid of perturbation theory or the renormalization-group method. It is shown that, in certain special cases, the exponents are Gaussian when

n

is a negative even integer and that

n=0

corresponds to the excluded-volume problem. The general case is shown to reduce to an arbitrary

m

-component model which is random when

n=0

and constrained when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. A direct derivation of the large-

n

limit is given and leads to a variety of exactly solvable models. Expressions for the order ${n}^{\ensuremath{-}1}$ correction are obtained in terms of correlation functions. This expansion is valid at all temperatures and for any order of transition, so that it is particularly suitable for considering tricritical phenomena.