Sergey Bravyi, IBM Watson Center

Robert Raussendorf, Perimeter Institute

Perugia July 16, 2007

Exactly solvable models of statistical physics: applications for

quantum computing

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: 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

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 outcomesof all earlier measurements

end do

for j=1 to n do

Measurement-based QC:

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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.

• 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:

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 statisticsof the measurement outcomes

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:

Hamiltonian:

The planar code state is the unique ground state of H

The planar code state is uniquely defined by equations

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

Duality between 1-cycles and cuts

1-cycle cut

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)

Linear spaces of cuts and 1-cycles are dual to each other:

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

Computing probabilities for complete measurements:

a cut

= Ising spin

- Probability of the outcome for a complete measurement (every qubit is measured)

Introduce local “temperature” :

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 clusterstate 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

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

Computing conditional probabilities

Measured qubits Unmeasured qubits

E

For any define a relative planar code state

Then

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 outcomefor incomplete measurement:

Intermediate result: MQC with the planar code state isclassically simulatable if at every step of MQC the setof measured qubits is simply connected

Disclaimer: the doubling trick works only if the set of measured qubits E is simply connected (no holes)

Extension to arbitrary measurement patterns:

E = measured qubits

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

If has more than one connected component,

Extension to arbitrary measurement patterns:

Suppose the doubled graph can be drawn ona surface of genus g. Then

is the Lagrangian subspace

Barahona’s 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

Pfaffian formula for planar graphs

is Kasteleyn orientation (a flux through any plaquette is 1)

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:

g = genus of the doubled graph obtained by gluingtogether two copies of E

The sum contains terms

can be efficiently computed if

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 WeightEnumerators, Knill & Laflamme

Evaluation of Jonespolynomials and TQFT invariants, Freedman et al.

Contraction of tensor networks, Markov & Shi

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).