Download presentation

Presentation is loading. Please wait.

Published byCullen Torres Modified over 3 years ago

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).

Similar presentations

OK

Pretty-Good Tomography Scott Aaronson MIT. Theres a problem… To do tomography on an entangled state of n qubits, we need exp(n) measurements Does this.

Pretty-Good Tomography Scott Aaronson MIT. Theres a problem… To do tomography on an entangled state of n qubits, we need exp(n) measurements Does this.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on sound navigation and ranging systematic Ppt on communication in hindi Ppt on challenges of democracy in india Ppt on drugs and alcohol abuse Ppt on marketing management by philip kotler definition Ppt on lok sabha election 2014 Ppt on development of dentition Ppt on consistent and inconsistent equations Male reproductive system anatomy and physiology ppt on cells Ppt on personality development slides