Alexei Kitaev

Entanglement in quantum critical phenomena

Entanglement, one of the most intriguing features of quantum theory and a main resource in quantum information science, is expected to play a crucial role also in the study of quantum phase transitions, where it is responsible for the appearance of long-range correlations. We investigate, through a microscopic calculation, the scaling properties of entanglement in spin chain systems, both near and at a quantum critical point. Our results establish a precise connection between concepts of quantum information, condensed matter physics, and quantum field theory, by showing that the behavior of critical entanglement in spin systems is analogous to that of entropy in conformal field theories. We explore some of the implications of this connection.

Anyons in an exactly solved model and beyond

A spin-1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 source gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and non-Abelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.

Topological entanglement entropy

We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal density operator rho for the degrees of freedom in the interior. The von Neumann entropy of rho, a measure of the entanglement of the interior and exterior variables, has the form S(rho) = alphaL - gamma + ..., where the ellipsis represents terms that vanish in the limit L --> infinity. We show that - gamma is a universal constant characterizing a global feature of the entanglement in the ground state. Using topological quantum field theory methods, we derive a formula for gamma in terms of properties of the superselection sectors of the medium.

Periodic table for topological insulators and superconductors

Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z, or Z_2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of K-homology. This classification is robust with respect to disorder, provided electron states near the Fermi energy are absent or localized. In some cases (e.g., integer quantum Hall systems) the K-theoretic classification is stable to interactions, but a counterexample is also given.

Topological quantum memory

We analyze surface codes, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An order-disorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the accuracy threshold), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a three-dimensional Z(2) lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are local, that qubits can be measured rapidly, and that polynomial-size classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however, for this procedure the quantum gates are local only if the qubits are arranged in four or more spatial dimensions. We discuss procedures for encoding, measurement, and performing fault-tolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.

Encoding a qubit in an oscillator

Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables q and p. In the setting of quantum optics, fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting; however, nonlinear mode coupling is required for the preparation of the encoded states. Finite-dimensional versions of these codes can be constructed that protect encoded quantum information against shifts in the amplitude or phase of a d-state system. Continuous-variable codes can be invoked to establish lower bounds on the quantum capacity of Gaussian quantum channels.

The Complexity of the Local Hamiltonian Problem

The k-LOCAL HAMILTONIAN problem is a natural complete problem for the complexity class QMA, the quantum analog of NP. It is similar in spirit to MAX-k-SAT, which is NP-complete for k > 2. It was known that the problem is QMA-complete for any k > 3. On the other hand 1-LOCAL HAMILTONIAN is in P, and hence not believed to be QMA-complete. The complexity of the 2-LOCAL HAMILTONIAN problem has long been outstanding. Here we settle the question and show that it is QMA-complete. We provide two independent proofs; our first proof uses a powerful technique for analyzing the sum of two Hamiltonians; this technique is based on perturbation theory and we believe that it might prove useful elsewhere. The second proof uses elementary linear algebra only. Using our techniques we also show that adiabatic computation with two-local interactions on qubits is equivalent to standard quantum computation.

Parallelization, amplification, and exponential time simulation of quantum interactive proof systems

In this paper we consider quantum interactive proof systems, which are interactive proof systems in which the prover and verifier may perform quantum computations and exchange quantum information. We prove that any polynomial-round quan tum interactive proof system with two-sided bounded error can be parallelized to a quantum interactive proof system with exponentially small one-sided error in which the prover and verifier exchange only 3 messages. This yields a simplified proof that PSPACE has 3-message quantum interactive proof systems. We also prove that any language having a quantum interactive proof system can be decided in deterministic exponential time, implying that single-prover quantum interactive proof systems are strictly less powerful than multiple-prover classical interactive proof systems unless EXP -NEXP. 1. I N T R O D U C T I O N Interactive proof systems were introduced by Babai [3] and Goldwasser, Micali, and Rackoff [17] in 1985. In the same year, Deutsch [10] gave the first formal t reatment of quantum computation. Since then, both subjects have received a great deal of at tention and have generated a number of exciting results, perhaps most notably the IP ---PSPACE characterization of Lund, Fortnow, Karloff, and Nisan [25] and Shamir [26], and the polynomial-time quan tum algorithms for factoring and discrete logarithms due to Shor [28]. In this paper we consider quantum interactive proof systems, which merge notions from these two subjects. A quantum interactive proof system consists of two par t ies -a prover with unbounded quantum computational power and a quantum polynomiai-time verifier--that communicate through a *Microsoft Research, One Microsoft Way, Redmond, WA 98052, e-mail: kitaev(~microsoft.com. On leave from L.D. Landau Insti tute for Theoretical Physics tDepar tment of Computer Science, University of Calgary, 2500 University Drive NW, Calgary (Alberta), Canada T2N 1N4, e-mail: jwatrous@cpsc.ucalgary.ca. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the lull citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. STOC 2000 Portland Oregon USA Copyright ACM 2000 1-58113 184-4/00/5...

Simulation of topological field theories by quantum computers

5.00 quantum channel. As in the case of classical interactive proof systems, the prover at tempts to prove to the verifier that a given input string satisfies some specified property, while the verifier tries to determine the validity of this proof. A language L is said. to have a quantum interactive proofsystem if there exists a quantum verifier V such that (i) there exists a quan tum prover P that can always convince V to accept when the input is in L, and (ii) no quan tum prover P can convince V to accept with nonnegligible probability when the input is not in L. Quantum interactive proof systems were first studied in a paper by one of us [30], wherein it was shown that every PSPACE language has a quantum interactive proof system, with exponentially small one-sided error, in which the prover and verifier exchange a total of only 3 messages. This implies that any classical interactive proof system can be parallelized to require just 3 messages in the quantum setting, which is a task that cannot be accomplished classically unless the polynomial-time hierarchy collapses to AM [3; 18]. In this paper we prove the following stronger result: any quantum interactive proof system can be parallelized to a 3-message quan tum protocol with exponentially small onesided error. In order to achieve exponentially small error in the 3-message case, we prove the somewhat surprising fact that entanglement among parallel repetitions of a 3message quantum interactive proof system gives a cheating prover absolutely no increase in success probability. Our result simplifies the proof that PSPACE has 3-message quantum interactive proof systems, in the sense that it treats any classical protocol for a given PSPACE language as a black-box. While (single-prover) classical interactive proof systems recognize precisely those languages in PSPACE, it was shown by Babai, Fortnow, and Lund [4] that any language in nondeterministic exponential time (NEXP) has a two-prover interactive proof system, wherein the two provers are not permitted to communicate with one another during the protocol. A sequence of papers [9; 13; 24] led to a result of Feige and Lov~sz [14] that any language in NEXP has a two-prover interactive proof system requiring just one round of communication (meaning that the verifier sends one question to each of the provers in parallel, then receives their responses). A natural question to ask is whether NEXP has single-prover quantum interactive proof systems, or equivalently whether single-prover quantum interactive proof systems can simulate multiple classical provers. We show that this is not likely to be the case, as any language having a quantum interactive proof system is necessarily contained in determin-

Topological phases of fermions in one dimension

Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold:1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”.2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.