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Transcript
Exactly solvable models of statistical physics: applications for quantum computing Sergey Bravyi, IBM Watson Center Robert Raussendorf, Perimeter Institute Perugia July 16, 2007 Outline • Measurement-based quantum computation (MQC) • Classical simulation of MQC • Kitaev’s toric code model and the planar code states • Reduction from MQC with the planar code states to the Ising model on a planar graph (doubling trick) • Barahona’s Pfaffian formula for planar and non-planar graphs Measurement-based QC: resource state • Step 1: 2: prepare measurenqubits qubit resource of the resource state state one by oneThe using projective measurement. resource statenon-destructive is algorithm-independent The measurement pattern is algorithm specific Example: cluster state (universal resource) Measurement-based QC: measurement pattern • Step 2 (algorithm specific): for j=1 to n do Measure qubit q(j) projectively using orthonormal basis The outcome is a random bit A choice of and q(j) may depend on the outcomes of all earlier measurements end do Measurement-based QC: 1 4 7 2 5 8 3 6 9 Measurement-based QC • Step 3: extract the answer by classical postprocessing of the random bit string Theorem (Briegel & Raussendorf 01) Any problem that can be solved on a quantum computer in polynomial time can be solved by MQC with the cluster state in polynomial time. Advantages of MQC: • Entangling operations = nearest neighbors Ising interactions • Noisy resource state can be efficiently purified • Can be made fault-tolerant with very high threshold in 3D Classical simulation of MQC Output of MQC is a random bit string with a probability distribution Classical simulator must be able to reproduce statistics of the measurement outcomes Definition: MQC is classically simulatible if there exists a classical algorithm with a running time poly(n) that computes conditional probabilities For which resource states MQC is classically simulatable? • Graph states with a treewidth (Markov & Shi 05). Includes 1D and quasi-1D cluster states • States with a entanglement width (Briegel, Vidal, et al. 06) Includes matrix product states • Our result: planar code states and surface code states of genus . These states have treewidth and entanglement width The planar code state: planar version of Kitaev’s toric code Plaquette operators: Vertex operators: The planar code state is uniquely defined by equations Hamiltonian: The planar code state is the unique ground state of H Planar code state = superposition of 1-cycles A basis vector = subset of edges labeled by ‘1’ = 1-chain is a set of 1-cycles on the lattice (a linear space mod 2) 1-cycle is a 1-chain that has even number of edges incident to every vertex Duality between 1-cycles and cuts 1-cycle cut of cuts and 1-cycles are dual to each other: A Linear 1-chainspaces y is called a cut iff one can color the set of vertices using blue and green colors such that every edge of y has blue and green endpoints Let be a set of all cuts on the lattice (a linear space mod 2) Duality between 1-cycles and cuts: Hadamard gate: Conclusion: the planar code state is a uniform superposition of all cuts on the lattice (after a local change of basis) The states and are equivalent for MQC Computing probabilities for complete measurements: - Probability of the outcome for a complete measurement (every qubit is measured) Introduce local “temperature” : a cut = Ising spin Computing probabilities for complete measurements: Barahona (1982): on a planar graph can be computed in time poly(n) for arbitrary (complex) weights 2D cluster state: computing the probabilities for complete measurements is quantum-NP hard Corollary: the planar code state can not be converted to the 2D cluster state by performing one-qubit measurements on a subset of qubits (even with exp. small success probability) Computing conditional probabilities Conclusion: we need to compute probabilities of incomplete measurements E is the subset of measured qubits and Incomplete overlap Computing conditional probabilities Measured qubits Unmeasured qubits E Relative 1-cycle Boundary Given a 1-chain a boundary such as that a set of vertices A relative 1-cyclexisdefine a 1-chain that are incident to odd number of edges from x = set of relative 1-cycles Computing conditional probabilities Measured qubits Unmeasured qubits E For any Then define a relative planar code state Computing conditional probabilities: doubling trick We need to compute an incomplete overlap: Key idea: the state is the planar code state for a planar graph obtained from two copies of E by identifying vertices of Computing conditional probabilities: doubling trick Now we can efficiently compute probability of any outcome for incomplete measurement: Disclaimer: the doubling trick works only if the set of measured qubits E is simply connected (no holes) Intermediate result: MQC with the planar code state is classically simulatable if at every step of MQC the set of measured qubits is simply connected Extension to arbitrary measurement patterns: Let x be a relative 1-cycle on E obtained by restricting a 1-cycle on the complete lattice to E has even number of vertices on every connected part of If E = measured qubits has more than one connected component, Extension to arbitrary measurement patterns: Suppose the doubled graph a surface of genus g. Then can be drawn on is the Lagrangian subspace Barahona’s reduction to the dimer model: G can be arbitrary graph The graph is obtained from by adding O(n) vertices and edges = set of dimer configurations Dimer configuration Pfaffian formula for planar graphs is Kasteleyn orientation (a any plaquette is 1) flux through Extension to arbitrary measurement patterns: Applying Barahona’s construction we get is a fixed dimer configuration is a 1-cycle Summation over spin structures Definition: Properties: Pfaffian formula for non-planar graphs (Cimasoni and Reshetikhin 07) is efficiently computable is Kasteleyn orientation associated with a spin structure f Extension to arbitrary measurement patterns: The sum contains terms g = genus of the doubled graph obtained by gluing together two copies of E can be efficiently computed if Simulating quantum computation on a classical computer: do we already know all cases when it is possible ? Quadratically Signed Weight Enumerators, Knill & Laflamme Evaluation of Jones polynomials and TQFT invariants, Freedman et al. Contraction of tensor networks, Markov & Shi Adiabatic evolution algorithm (simulated annealing), Farhi et al. Quantum walks (diffusion), Ambainis et al. Simulation of “fermionic linear optics” Valiant, DiVincenzo et al. Main goal: find a family of quantum algorithms that can be efficiently simulated classically via a mapping to exactly solvable models of statistical physics (we shall consider the Ising model on planar and “almost planar” graphs).