Sergey Bravyi, IBM Watson Center Robert Raussendorf, Perimeter Institute Perugia July 16, 2007 Exactly solvable models of statistical physics: applications for quantum computing
Mar 31, 2015
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).