Exactly solvable models of statistical physics: applications for quantum computing

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

2. 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

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

4. 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 outcomesof all earlier measurements end do

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

6. Measurement-based QC • Step 3: extract the answer by classical postprocessingof the random bit string Theorem(Briegel & Raussendorf 01)Any problem that can be solved on a quantum computer inpolynomial time can be solved by MQC with the cluster statein 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

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

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 ofgenus . These states have treewidth andentanglement width

9. 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

10. 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

11. Duality between 1-cycles and cuts 1-cycle cut Linear spaces of cuts and 1-cycles are dual to each other: A 1-chain y is called a cut iff one can color the set of verticesusing blue and green colors such that every edge of y has blueand green endpoints Let be a set of all cuts on the lattice (a linear space mod 2)

12. Duality between 1-cycles and cuts: Hadamard gate: Conclusion:the planar code state is a uniform superpositionof all cuts on the lattice (after a local change of basis) The states and are equivalent for MQC

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

14. Computing probabilities for complete measurements: Barahona (1982): on a planar graph can becomputed 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 clusterstate 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 qubits Unmeasured qubits E Relative 1-cycle Boundary Given a 1-chain x define a boundary as a set of vertices that are incident to odd number of edges from x A relative 1-cycle is a 1-chain such that = set of relative 1-cycles

17. Computing conditional probabilities Unmeasured qubits Measured 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 outcomefor 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 isclassically simulatable if at every step of MQC the setof measured qubits is simply connected

20. Extension to arbitrary measurement patterns: Let x be a relative 1-cycle on Eobtained by restricting a 1-cycleon the complete lattice to E has even number of verticeson every connected part of E = measured qubits If has more than one connected component,

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

22. Barahona’s reduction to the dimer model: G can be arbitrary graph The graph is obtained from byaddingO(n)vertices and edges Dimer configuration = 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 Barahona’s 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: The sum contains terms g = genus of the doubled graph obtained by gluingtogether two copies of E can be efficiently computed if

28. Simulating quantum computation on a classical computer: do we already know all cases when it is possible ? Quadratically Signed WeightEnumerators, Knill & Laflamme Adiabatic evolution algorithm (simulated annealing), Farhi et al. Evaluation of Jonespolynomials and TQFT invariants, Freedman et al. Quantum walks (diffusion), Ambainis et al. Contraction of tensor networks, Markov & Shi Simulation of “fermionic linear optics”Valiant, DiVincenzo et al. Main goal: find a family of quantum algorithms that can be efficiently simulatedclassically via a mapping to exactly solvable models of statistical physics (we shallconsider the Ising model on planar and “almost planar” graphs).