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arXiv:quant-ph/0309064 · New Journal of Physics

On the quantum computational complexity of the Ising spin glass partition function and of knot invariants

Daniel A Lidar

Published Nov 17, 2004• 33 citations
arXivHTMLPDF

Abstract

It is shown that the canonical problem of classical statistical thermodynamics, the computation of the partition function, is in the case of ± J Ising spin glasses a particular instance of certain simple sums known as quadratically signed weight enumerators (QWGTs). On the other hand, it is known that quantum computing is polynomially equivalent to classical probabilistic computing with an oracle for estimating certain QWGTs. This suggests a connection between the partition function estimation problem for spin glasses and quantum computation. This connection extends to knots and graph theory via the equivalence of the Kauffman bracket polynomial and the partition function for the Potts model.

DOI
10.1088/1367-2630/6/1/167

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It is shown that the canonical problem of classical statistical thermodynamics, the computation of the partition function, is in the case of ± J Ising spin glasses a particular instance of certain simple sums known as quadratically signed weight enumerators (QWGTs). On the other hand, it is known that quantum computing is polynomially equivalent to classical probabilistic computing with an oracle for estimating certain QWGTs. This suggests a connection between the partition function estimation problem for spin glasses and quantum computation. This connection extends to knots and graph theory via the equivalence of the Kauffman bracket polynomial and the partition function for the Potts model.
It is shown that the canonical problem of classical statistical thermodynamics, the computation of the partition function, is in the case of +/-J Ising spin glasses a particular instance of certain simple sums known as quadratically signed weight enumerators (QWGTs). On the other hand it is known that quantum computing is polynomially equivalent to classical probabilistic computing with an oracle for estimating QWGTs. This suggests a connection between the partition function estimation problem for spin glasses and quantum computation. This connection extends to knots and graph theory via the equivalence of the Kauffman polynomial and the partition function for the Potts model.
We provide a quantum algorithm for the exact evaluation of the Potts partition function for a certain class of restricted instances of graphs that correspond to irreducible cyclic codes. We use the same approach to demonstrate that quantum computers can provide an exponential speed up over the best classical algorithms for the exact evaluation of the weight enumerator polynomial for a family of classical cyclic codes. In addition to this we also provide an efficient quantum approximation algorithm for a function (signed-Euler generating function) closely related to the Ising partition function and demonstrate that this problem is BQP-complete. \n\nWe accomplish the above for the Potts partition function by using a series of links between Gauss sums, classical coding theory, graph theory and the partition function. We exploit the fact that there exists an efficient approximation algorithm for Gauss sums and the fact that this problem is equivalent in complexity to evaluating discrete log. A theorem of McEliece allows one to turn the Gauss sum approximation into an exact evaluation of the Potts partition function. Stripping the physics from this result leaves one with the result for the weight enumerator polynomial. \n\nThe result for the approximation of the signed-Euler generating function was accomplished by fashioning a new mapping between quantum circuits and graphs. The mapping provided us with a way of relating the cycle structure of graphs with quantum circuits. Using a slight variant of this mapping, we present the final result of this thesis which presents a way of testing families of quantum circuits for their classical simulatability. We thus provide an efficient way of deciding whether a quantum circuit provides any additional computational power over classical computation and this is achieved by exploiting the fact that planar instances of the Ising partition function (with no external magnetic field) can be efficiently classically computed.
We prove that quantum computation is polynomially equivalent to classical probabilistic computation with an oracle for estimating the value of simple sums, quadratically signed weight enumerators. The problem of estimating these sums can be cast in terms of promise problems and has two interesting variants. An oracle for the unconstrained variant may be more powerful than quantum computation, while an oracle for a more constrained variant is efficiently solvable in the one-bit model of quantum computation. Thus, problems involving estimation of quadratically signed weight enumerators yield problems in BQP (bounded error quantum polynomial time) that are distinct from the ones studied so far, include a canonical BQP complete problem, and can be used to define and study complexity classes and their relationships to quantum computation.
We exploit a recently constructed mapping between quantum circuits and graphs in order to prove that circuits corresponding to certain planar graphs can be efficiently simulated classically. The proof uses an expression for the Ising model partition function in terms of quadratically signed weight enumerators (QWGTs), which are polynomials that arise naturally in an expansion of quantum circuits in terms of rotations involving Pauli matrices. We combine this expression with a known efficient classical algorithm for the Ising partition function of any planar graph in the absence of an external magnetic field, and the Robertson–Seymour theorem from graph theory. We give as an example a set of quantum circuits with a small number of non-nearest-neighbor gates which admit an efficient classical simulation.
We present a polynomial time algorithm for the construction of the Gibbs distribution of configurations in the Ising model, on a quantum computer. The algorithm is designed so that each run provides one configuration with exactly the correct quantum probability, namely, the corresponding thermodynamic weight. The partition function is thus approximated in the most efficient way. The algorithm does not suffer from critical slowing down, unlike its classical counterparts. Moreover, the algorithm can be applied in any dimension, to any of the frustrated Ising models, diluted Ising models, and spin glasses. In particular, we show that under certain plausible assumptions concerning the scaling of the number of low-lying metastable states, the ground state of any dimensional short-range spin glass can be found in poly(N_T) steps, N_T being the number of spins (N) in the system plus the number of bonds between them (N_B). Since the problem of finding the ground state of a spin glass in 3D is known to be NP-complete, we thus provide evidence that possibly P=NP on a quantum computer.
We prove that the 2D Ising model is complete in the sense that the partition function of any classical q-state spin model (on an arbitrary graph) can be expressed as a special instance of the partition function of a 2D Ising model with complex inhomogeneous couplings and external fields. In the case where the original model is an Ising or Potts-type model, we find that the corresponding 2D square lattice requires only polynomially more spins with respect to the original one, and we give a constructive method to map such models to the 2D Ising model. For more general models the overhead in system size may be exponential. The results are established by connecting classical spin models with measurement-based quantum computation and invoking the universality of the 2D cluster states.