Quantum Algorithms Lecture #3. Stephen Jordan

Transcription

1 Quantum Algorithms Lecture #3 Stephen Jordan

2 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

3 Summary of Lecture 2 Introduced more building blocks Used these blocks to build quantum algorithms: Oracles & Recursion Hadamard Test Hadamard Transform Fourier Transform Deutsch-Jozsa Algorithm Bernstein-Vazirani Algorithm Shor's Algorithm Introduced Hidden Subgroup and Hidden Shift

4 This Time Quantum Algorithms for Topological Invariants knot invariants 3-manifold invariants BQP-hardness DQC1-hardness

5 Knot Theory A knot is an embedding of the circle into. Knots are considered equivalent if one can be deformed into another without cutting.

6 Knot Equivalence

7 Links A link is an embedding of an arbitrary number of circles into. unlink of two strands Hopf link Borromean rings

8 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.

9 Reidemeister Moves Diagrams of equivalent knots are always reachable by some sequence of the three Reidemeister moves. Move 1: Move 2: Move 3:

10 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.

11 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 different values then it is a complete invariant.

12 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.

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14 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.

15 Jones Polynomial as Physics 1985: Jones discovers the Jones Polynomial 1989: Witten discovers Jones polynomial arises as an amplitude in Chern-Simons Theory Church-Turing-Deutsch Thesis: Every physically realizable computation can be simulated by quantum circuits with polynomial overhead. 2000: Freedman, Kitaev, Larsen, and Wang find quantum algorithm and hardness result for Jones polynomials.

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

17 Non-Abelian Anyons Two-dimensional condensed-matter systems may have anyonic quasiparticle excitations. Braiding can induce unitary transformations within degenerate ground space.

18 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

19 The Braid Group

20 The Artin Generators Artin's theorem: these relations capture all topological equivalences of braids.

21 Commutation

22 Yang-Baxter Equation

23 A braid: Its plat closure: Its trace closure:

24 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

25 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. Exercise: prove it. trace plat

26 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:

27 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:

28 Markov Moves Move 2:

29 Link Invariants from Braid Group Representations Let be a family of representations of the braid groups is automatically invariant under the first Markov move.

30 The Jones Representation The Jones polynomial is a polynomial in t. Coefficients c,d are functions of t. If t is a root of unity, then the representation is unitary.

31 The Jones Representation This example is for. In general the labels can take values beyond 0,1. The Jones polynomial is a polynomial in t. Coefficients c,d are functions of t. If t is a root of unity, then the representation is unitary.

32 Quantum Algorithm for Jones 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.

33 Choose random x:

34 Choose random x:

35 Approximating Jones Polynomials The preceeding algorithm achieves an additive approximation to the Jones polynomial. By sampling If times one obtains: is a random element of then:

36 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.

37 Plat Closures The Jones polynomial of this knot is. The Jone polynomial of this knot is.

38 O(n) samples yields:

39 The Jone polynomial of this knot is. Given a quantum circuit of G gates on n qubits, one can efficiently find a braid of poly(g) crossings on poly(n) strands such that: If we could additively approximate the Jones polynomial of a plat closure classically then we could simulate all of quantum computation!

40 The Jone polynomial of this knot is. Given a quantum circuit of G gates on n qubits, one can efficiently find a braid of poly(g) crossings on poly(n) strands such that: Additively approximating the Jones polynomial of a plat closre to 1/poly(n) precision is BQP-complete.

41 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 be constructed as either braid or plat closure.

42 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. These problems differ in precision.??? Any knot can be constructed as either braid or plat closure.

43 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.

44 DQC1 The class of problems solvable with one clean qubit is called DQC1. Loosely corresponds to NMR computers. Probably looks like this: If so, estimating the Jones polynomial of the trace closure of braids to polynomial additive precision is classically intractable for hardest instances.

45 3-Manifold Equivalence Three manifold: topological space locally like Homeomorphism: continuous map with continuous inverse. Fundamental question: given two manifolds, are they homeomorphic ( the same ).

46 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.

47 Two triangulations yield equivalent 3-manifolds iff they are connected by a finite sequence of Pachner moves.

48 Two triangulations yield equivalent 2-manifolds iff they are connected by a finite sequence of Pachner moves.

49 Pachner's Theorem Two triangulations yield equivalent n-manifolds iff they are connected by a finite sequence of Pachner moves. The Pachner moves correspond to the ways of gluing together n-simplices to obtain an (n+1)-simplex.

50 complexity of topological equivalence problems: knots in 2-manifolds in P 3-manifolds computable 4-manifolds uncomputable partial solution: manifold invariant if manifolds A and B are homeomorphic then f(a) = f(b)

51 Constructing Invariants To each tetrahedron associate a 6-index tensor. For each glued face, contract (sum over) the corresponding indices. We just need to find a tensor such that this sum is invariant under the Pachner moves.

52 Constructing Invariants To each tetrahedron associate a 6-index tensor. 6j tensor from : Ponzano-Regge 6j tensor from : Turaev-Viro

53 A TQFT maps n-manifolds to vector spaces and (n+1)-manifolds to linear maps between the vector spaces of its boundaries.

54 Functorial property: the gluing of two manifolds yields the composition of the associated linear maps.

55 Exercise Q. Argue that: is a projector

56 Exercise Q. Argue that: is a projector A. glue = functor

57 Exercise The empty boundary corresponds to is a projector is a vector is a dual vector Closed manifolds map to scalars..

58 Spin Foam Gravity Boundary is triangulated surface with labeled edges. These specify the geometry of space. The value of the tensor network is the transition amplitude between geometries.

59 Every physical system can be efficiently simulated by a standard quantum computer. t We should be able to estimate this amplitude with an efficient quantum circuit.

60 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.

61 Heegaard Splitting Specify: genus g gluing map between genus-g surfaces Every 3-manifold can be obtained this way.

62 Small changes to gluing map don't affect topology of resulting 3-manifold. It suffices to specify gluing map modulo those small changes. Result is element of mapping class group. Fairly intuitive: generated by Dehn twists:

63 Alternative Definition of TV invariant t genus g handlebody mapping class group element t genus g handlebody is a unitary representation of MCG

64 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.

65 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 TV even though approximation is trivial-on-average!

66 BQP-complete DQC1-complete

67 Ponzano-Regge Invariant Ponzano-Regge invariant can be efficiently approximated on a quantum computer. Actually only need a permutational computer. Probably an easier problem than estimating Turaev-Viro invariant. Turaev-Viro: 6j tensor from Ponzano-Regge: 6j tensor from

68 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

69 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.

70 Further Reading Surveys: Childs & van Dam, Quantum algorithms for algebraic problems [arxiv: ] Mosca, Quantum algorithms [arxiv: ] Jordan, Quantum algorithm zoo math.nist.gov/quantum/zoo/ Childs, lecture notes ~amchilds/teaching/w13/qic823.html