1. Sergey Bravyi, IBM Watson Center Robert Raussendorf, Perimeter Institute Perugia July 16, 2007 Exactly solvable models of statistical physics: applications for quantum computing

2. Outline Measurement-based quantum computation (MQC) Classical simulation of MQC Kitaevs 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) Barahonas Pfaffian formula for planar and non-planar graphs

3. Measurement-based QC: resource state Step 1: prepare n qubit resource state The resource state is algorithm-independent Example: cluster state (universal resource) Step 2: measure qubits of the resource state one by one using projective non-destructive measurement. The measurement pattern is algorithm specific

4. Measurement-based QC: measurement pattern Step 2 (algorithm specific): 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 for j=1 to n do

5. Measurement-based QC: 1 2 3 4 5 6 7 8 9

6. 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. Entangling operations = nearest neighbors Ising interactions Noisy resource state can be efficiently purified Can be made fault-tolerant with very high threshold in 3D Advantages of MQC:

7. Classical simulation of MQC Output of MQC is a random bit string with a probability distribution MQC is classically simulatible if there exists a classical algorithm with a running time poly(n) that computes conditional probabilities Definition: Classical simulator must be able to reproduce statistics of the measurement outcomes

8. 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

9. The planar code state: planar version of Kitaevs toric code Plaquette operators: Vertex operators: Hamiltonian: The planar code state is the unique ground state of H The planar code state is uniquely defined by equations

10. Planar code state = superposition of 1-cycles 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 A basis vector = subset of edges labeled by 1 = 1-chain

11. Duality between 1-cycles and cuts 1-cyclecut A 1-chain 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) Linear spaces of cuts and 1-cycles are dual to each other:

12. 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

13. Computing probabilities for complete measurements: a cut = Ising spin - Probability of the outcome for a complete measurement (every qubit is measured) Introduce local temperature :

14. 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)

15. Computing conditional probabilities Conclusion: we need to compute probabilities of incomplete measurements E is the subset of measured qubits and Incomplete overlap

16. Computing conditional probabilities Measured qubitsUnmeasured qubits E Boundary A relative 1-cycle is a 1-chain such that = set of relative 1-cycles Relative 1-cycle Given a 1-chain x define a boundary as a set of vertices that are incident to odd number of edges from x

17. Computing conditional probabilities Measured qubits Unmeasured qubits E For any define a relative planar code state Then

18. 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

19. Computing conditional probabilities: doubling trick Now we can efficiently compute probability of any outcome for incomplete measurement: 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 Disclaimer: the doubling trick works only if the set of measured qubits E is simply connected (no holes)

20. Extension to arbitrary measurement patterns: E = measured qubits 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 has more than one connected component,

21. Extension to arbitrary measurement patterns: Suppose the doubled graph can be drawn on a surface of genus g. Then is the Lagrangian subspace

22. Barahonas reduction to the dimer model: Dimer configuration G can be arbitrary graph The graph is obtained from by adding O(n) vertices and edges = set of dimer configurations

23. Pfaffian formula for planar graphs is Kasteleyn orientation (a flux through any plaquette is 1)

24. Extension to arbitrary measurement patterns: Applying Barahonas construction we get is a fixed dimer configuration is a 1-cycle

25. Summation over spin structures Definition: Properties:

26. Pfaffian formula for non-planar graphs (Cimasoni and Reshetikhin 07) is efficiently computable is Kasteleyn orientation associated with a spin structure f

27. Extension to arbitrary measurement patterns: g = genus of the doubled graph obtained by gluing together two copies of E The sum contains terms can be efficiently computed if

28. Simulating quantum computation on a classical computer: do we already know all cases when it is possible ? Adiabatic evolution algorithm (simulated annealing), Farhi et al. Quantum walks (diffusion), Ambainis et al. Simulation of fermionic linear optics Valiant, DiVincenzo et al. Quadratically Signed Weight Enumerators, Knill & Laflamme Evaluation of Jones polynomials and TQFT invariants, Freedman et al. Contraction of tensor networks, Markov & Shi 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).