Quantum AlgorithmsLecture #3

Stephen Jordan

Summary of Lecture 1

● Defined quantum circuit model.● Argued it captures all of quantum computation.● Developed some building blocks:

– Gate universality– Controlled-unitaries– Reversible computing– Phase kickback– Phase estimation

Summary of Lecture 2● Introduced more building blocks

– Oracles & Recursion– Hadamard Test– Hadamard Transform– Fourier Transform

● Used these blocks to build quantum algorithms:– Deutsch-Jozsa Algorithm– Bernstein-Vazirani Algorithm– Shor's Algorithm

● Introduced Hidden Subgroup and Hidden Shift

This Time

● Quantum Algorithms for Topological Invariants– knot invariants– 3-manifold invariants– BQP-hardness– DQC1-hardness

Knot Theory

A knot is an embedding of the circle into .

Knots are considered equivalent if one can bedeformed into another without cutting.

Knot Equivalence

LinksA link is an embedding of an arbitrary number of circles into .

unlink of two strands

Hopf link

Borromean rings

Knot Equivalence Problem

Given two knots, decide equivalence.

Knots can be specified by knot diagrams, which are degree-4 graphs with each vertex labeled as either or .

Reidemeister MovesDiagrams of equivalent knots are always reachable by some sequence of the three Reidemeister moves.

Move 1:

Move 2:

Move 3:

Unknot Problem

Unknot problem: decide whether a given knot is equivalent to the unknot.

● UNKNOT NP. [Lagarias & Pippenger, 1999]

● UNKNOT coNP (assuming GRH).[Kuperberg, 2011]

● UNKNOT is not known to be in P.

Knot Invariants

Lacking an algorithm for UNKNOT, one can make partial progress with knot invariants.

If are equivalent knots then .

If f always maps inequivalent knots to differentvalues then it is a complete invariant.

Jones Polynomial

The Jones polynomial is an invariant maps oriented links to polynomials in a single-variable.

The Jones polynomial is a strong invariant, but is known not to be complete for links.

Jones Polynomial

● The degree of the Jones polynomial is linear in the number of crossings in the knot diagram.(Thus it can be written down efficiently.)

● However, the coefficients can be exponentially large, and hard to compute.

● Exact computation of the Jones polynomial is #P-hard.

Jones Polynomial as Physics● 1985: Jones discovers the Jones Polynomial● 1989: Witten discovers Jones polynomial arises as

an amplitude in Chern-Simons Theory

● 2000: Freedman, Kitaev, Larsen, and Wang find quantum algorithm and hardness result for Jones polynomials.

Church-Turing-Deutsch Thesis:

Every physically realizable computation can be simulated by quantum circuits with polynomial overhead.

Anyons

● In (2+1)-D winding number is well-defined● Particle exchange can induce phase

Non-Abelian Anyons● Two-dimensional condensed-matter systems

may have anyonic quasiparticle excitations.

● Braiding can induce unitary transformations within degenerate ground space.

Non-Abelian Anyons

● Non-abelian anyons give us a unitary representation of the braid group.

● In some cases the set of unitary transformations induced by elementary crossings is a universal set of quantum gates.

“topological quantum computation”

The Braid Group

The Artin Generators

Artin's theorem: these relations capture alltopological equivalences of braids.

Commutation

Yang-Baxter Equation

A braid:

Its plat closure:

Its trace closure:

Alexander's Theorem: Any link can be obtained as the trace closure of some braid.

Corollary: Any link can be obtained as the plat closure of some braid.

trace plat

Alexander's Theorem: Any link can be obtained as the trace closure of some braid.

Corollary: Any link can be obtained as the plat closure of some braid.

trace plat

Exercise: prove it.

Alexander's Theorem: Any link can be obtained as the trace closure of some braid.

Corollary: Any link can be obtained as the plat closure of some braid.

Proof:

Markov Moves

A function on braids is an invariant of the corresponding trace closures if it is invariant under the two Markov moves.

Move 1:

Move 2:

Markov Moves

Move 2:

Link Invariants from Braid Group Representations

● Let be a family of representations of the braid groups

● is automatically invariant under the first Markov move.

The Jones Representation

The Jones polynomial is apolynomial in t.

Coefficients c,d are functions of t.

If t is a root of unity, then therepresentation is unitary.

The Jones Representation

The Jones polynomial is apolynomial in t.

Coefficients c,d are functions of t.

If t is a root of unity, then therepresentation is unitary.

This example is for .

In general the labels can takevalues beyond 0,1.

● We can implement the Jones representation by invoking gate universality.

● Church-Turing-Deutsch principle says this is natural - not just a lucky coincidence.

● We can estimate any diagonal matrix element of the representation using the Hadamard test.

Quantum Algorithm for Jones

Choose random x:

Choose random x:

Approximating Jones Polynomials

The preceeding algorithm achieves an additive approximation to the Jones polynomial.

By sampling times one obtains:

If is a random element of then:

Approximating Jones Polynomials

Our algorithm yields:

Is this nontrivial?

Probably: Arbitrary quantum circuits can be approximated by braids. Thus, we can estimate the trace of quantum circuits. This is “DQC1-complete”.

Also: we can do better.

Plat Closures

The Jones polynomial of thisknot is .

The Jone polynomial of thisknot is .

O(n) samples yields:

The Jone polynomial of thisknot is .

Given a quantum circuit of G gates on n qubits,one can efficiently find a braid of poly(G) crossingson poly(n) strands such that:

If we could additively approximate the Jonespolynomial of a plat closure classically then wecould simulate all of quantum computation!

The Jone polynomial of thisknot is .

Given a quantum circuit of G gates on n qubits,one can efficiently find a braid of poly(G) crossingson poly(n) strands such that:

Additively approximating the Jones polynomialof a plat closre to 1/poly(n) precision isBQP-complete.

● Additively approximating the Jones polynomial of the plat closures of braids is BQP-complete.

● Additively approximating the Jones polynomial of the trace closures braids is BQP-complete.

Any knot can beconstructed aseither braid orplat closure.

? ? ?

● Additively approximating the Jones polynomial of the plat closures of braids is BQP-complete.

● Additively approximating the Jones polynomial of the trace closures braids is BQP-complete.

Any knot can beconstructed aseither braid orplat closure.

? ? ?

These problems differ in precision.

One Clean Qubit

● One qubit starts in a pure state.

● n qubits are maximally mixed.

● Apply a polynomial size quantum circuit.

● Measure the first qubit.

DQC1● The class of problems solvable with oneclean qubit is called DQC1.

● Loosely corresponds to NMR computers.

Probably looks like this:

If so, estimating the Jones polynomial of the trace closureof braids to polynomial additive precision is classicallyintractable for hardest instances.

3-Manifold Equivalence

Three manifold: topological space locally like

Homeomorphism: continuous map withcontinuous inverse.

Fundamental question: given two manifolds, arethey homeomorphic (“the same”).

How do we describe a 3-manifold to a computer?

One way is to use a triangulation:

A set of tetrahedra.

A gluing of the faces.

Two triangulations yield equivalent 3-manifoldsiff they are connected by a finite sequence ofPachner moves.

Two triangulations yield equivalent 2-manifoldsiff they are connected by a finite sequence ofPachner moves.

Pachner's Theorem

Two triangulations yield equivalent n-manifoldsiff they are connected by a finite sequence ofPachner moves.

The Pachner moves correspond to the waysof gluing together n-simplices to obtain an(n+1)-simplex.

complexity of topological equivalence problems:

partial solution: manifold invariant – if manifolds A and B are homeomorphic then f(A) = f(B)

2-manifolds

3-manifolds

4-manifolds

knots

in P

computable

uncomputable

in

Constructing Invariants

To each tetrahedron associate a 6-index tensor.

For each glued face, contract (sum over) thecorresponding indices.

We “just” need to find a tensor such that this sumis invariant under the Pachner moves.

Constructing Invariants

To each tetrahedron associate a 6-index tensor.

6j tensor from : Ponzano-Regge

6j tensor from : Turaev-Viro

A TQFT maps n-manifolds to vector spacesand (n+1)-manifolds to linear maps betweenthe vector spaces of its boundaries.

Functorial property: the gluing of two manifoldsyields the composition of the associated linear maps.

Exercise

Q. Argue that:

is a projector

Exercise

Q. Argue that:

is a projector

A.

=

glue

functor

Exercise

The empty boundary corresponds to .

is a projector

is a vector

is a dual vector

Closed manifolds map to scalars.

Spin Foam Gravity

Boundary is triangulated surface with labelededges.

These specify the geometry of space.

The value of the tensor network is the transitionamplitude between geometries.

Every physical system can beefficiently simulated by astandard quantum computer.

We should be able to estimate this amplitudewith an efficient quantum circuit.

t

Turaev-Viro

● Additively estimating the Turaev-Viro invariant is a BQP-complete problem.– There is an efficient quantum algorithm.– Simulating a quantum computer reduces to

estimating the Turaev-Viro Invariant.

● The easiest proof is by reformulating the problem in terms of Heegaard splittings.

Heegaard Splitting

● Specify:– genus g– “gluing” map between genus-g surfaces

● Every 3-manifold can be obtained this way.

Small changes to gluing map don't affecttopology of resulting 3-manifold.

It suffices to specify gluing map modulothose small changes.

Result is element of mapping class group.

Fairly intuitive: generated by Dehn twists:

Alternative Definition of TV invariant

genus ghandlebody

genus ghandlebody

mapping classgroup element t

is a unitary representation of MCG

t

Quantum Algorithm for TV Invariant

Problem size: genus g and number of Dehn twists n

Algorithm:– Build from standard state using poly(g)

gates– Approximate unitary using polylog(g,n) gates– Estimate using Hadamard test.

TV Invariant is BQP-hard

● The images of the Dehn twist generators are like universal quantum gates.– They act on O(1) local degrees of freedom.– They generate a dense subgroup of the unitaries.

● Simulate a quantum circuit by translating each gate into a corresponding sequence of Dehn twists.

● Implies nontriviality of quantum algorithm for TVeven though approximation is trivial-on-average!

BQP-complete DQC1-complete

Ponzano-Regge Invariant

Turaev-Viro: 6j tensor from

Ponzano-Regge: 6j tensor from

Ponzano-Regge invariant can be efficientlyapproximated on a quantum computer.

Actually only need a permutational computer.

Probably an easier problem than estimatingTuraev-Viro invariant.

● Approximate Ponzano-Regge as follows:– Prepare a state of spin-1/2 particles with definite

total angular momentum– Permute them around– Measure total angular momentum of subsets

● “permutational computation”– Analogous to topological computation, but doesn't

need anyons– Dual to Aaronson's boson-sampling

I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

Further Reading

Surveys:● Childs & van Dam, Quantum

algorithms for algebraic problems [arXiv:0812.0380]

● Mosca, Quantum algorithms[arXiv:0808.0369]

● Jordan, Quantum algorithm zoomath.nist.gov/quantum/zoo/

● Childs, lecture noteshttp://www.math.uwaterloo.ca/~amchilds/teaching/w13/qic823.html