Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf ·...
Transcript of Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf ·...
![Page 1: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/1.jpg)
Quantum AlgorithmsLecture #3
Stephen Jordan
![Page 2: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/2.jpg)
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
![Page 3: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/3.jpg)
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
![Page 4: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/4.jpg)
This Time
● Quantum Algorithms for Topological Invariants– knot invariants– 3-manifold invariants– BQP-hardness– DQC1-hardness
![Page 5: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/5.jpg)
Knot Theory
A knot is an embedding of the circle into .
Knots are considered equivalent if one can bedeformed into another without cutting.
![Page 6: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/6.jpg)
Knot Equivalence
![Page 7: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/7.jpg)
LinksA link is an embedding of an arbitrary number of circles into .
unlink of two strands
Hopf link
Borromean rings
![Page 8: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/8.jpg)
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 .
![Page 9: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/9.jpg)
Reidemeister MovesDiagrams of equivalent knots are always reachable by some sequence of the three Reidemeister moves.
Move 1:
Move 2:
Move 3:
![Page 10: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/10.jpg)
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.
![Page 11: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/11.jpg)
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.
![Page 12: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/12.jpg)
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.
![Page 13: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/13.jpg)
![Page 14: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/14.jpg)
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.
![Page 15: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/15.jpg)
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.
![Page 16: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/16.jpg)
Anyons
● In (2+1)-D winding number is well-defined● Particle exchange can induce phase
![Page 17: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/17.jpg)
Non-Abelian Anyons● Two-dimensional condensed-matter systems
may have anyonic quasiparticle excitations.
● Braiding can induce unitary transformations within degenerate ground space.
![Page 18: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/18.jpg)
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”
![Page 19: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/19.jpg)
The Braid Group
![Page 20: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/20.jpg)
The Artin Generators
Artin's theorem: these relations capture alltopological equivalences of braids.
![Page 21: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/21.jpg)
Commutation
![Page 22: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/22.jpg)
Yang-Baxter Equation
![Page 23: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/23.jpg)
A braid:
Its plat closure:
Its trace closure:
![Page 24: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/24.jpg)
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
![Page 25: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/25.jpg)
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.
![Page 26: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/26.jpg)
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:
![Page 27: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/27.jpg)
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:
![Page 28: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/28.jpg)
Markov Moves
Move 2:
![Page 29: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/29.jpg)
Link Invariants from Braid Group Representations
● Let be a family of representations of the braid groups
● is automatically invariant under the first Markov move.
![Page 30: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/30.jpg)
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.
![Page 31: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/31.jpg)
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.
![Page 32: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/32.jpg)
● 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
![Page 33: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/33.jpg)
Choose random x:
![Page 34: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/34.jpg)
Choose random x:
![Page 35: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/35.jpg)
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:
![Page 36: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/36.jpg)
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.
![Page 37: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/37.jpg)
Plat Closures
The Jones polynomial of thisknot is .
The Jone polynomial of thisknot is .
![Page 38: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/38.jpg)
O(n) samples yields:
![Page 39: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/39.jpg)
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!
![Page 40: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/40.jpg)
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.
![Page 41: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/41.jpg)
● 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.
? ? ?
![Page 42: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/42.jpg)
● 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.
![Page 43: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/43.jpg)
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.
![Page 44: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/44.jpg)
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.
![Page 45: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/45.jpg)
3-Manifold Equivalence
Three manifold: topological space locally like
Homeomorphism: continuous map withcontinuous inverse.
Fundamental question: given two manifolds, arethey homeomorphic (“the same”).
![Page 46: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/46.jpg)
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.
![Page 47: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/47.jpg)
Two triangulations yield equivalent 3-manifoldsiff they are connected by a finite sequence ofPachner moves.
![Page 48: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/48.jpg)
Two triangulations yield equivalent 2-manifoldsiff they are connected by a finite sequence ofPachner moves.
![Page 49: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/49.jpg)
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.
![Page 50: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/50.jpg)
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
![Page 51: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/51.jpg)
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.
![Page 52: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/52.jpg)
Constructing Invariants
To each tetrahedron associate a 6-index tensor.
6j tensor from : Ponzano-Regge
6j tensor from : Turaev-Viro
![Page 53: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/53.jpg)
A TQFT maps n-manifolds to vector spacesand (n+1)-manifolds to linear maps betweenthe vector spaces of its boundaries.
![Page 54: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/54.jpg)
Functorial property: the gluing of two manifoldsyields the composition of the associated linear maps.
![Page 55: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/55.jpg)
Exercise
Q. Argue that:
is a projector
![Page 56: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/56.jpg)
Exercise
Q. Argue that:
is a projector
A.
=
glue
functor
![Page 57: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/57.jpg)
Exercise
The empty boundary corresponds to .
is a projector
is a vector
is a dual vector
Closed manifolds map to scalars.
![Page 58: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/58.jpg)
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.
![Page 59: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/59.jpg)
Every physical system can beefficiently simulated by astandard quantum computer.
We should be able to estimate this amplitudewith an efficient quantum circuit.
t
![Page 60: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/60.jpg)
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.
![Page 61: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/61.jpg)
Heegaard Splitting
● Specify:– genus g– “gluing” map between genus-g surfaces
● Every 3-manifold can be obtained this way.
![Page 62: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/62.jpg)
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:
![Page 63: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/63.jpg)
Alternative Definition of TV invariant
genus ghandlebody
genus ghandlebody
mapping classgroup element t
is a unitary representation of MCG
t
![Page 64: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/64.jpg)
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.
![Page 65: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/65.jpg)
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!
![Page 66: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/66.jpg)
BQP-complete DQC1-complete
![Page 67: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/67.jpg)
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.
![Page 68: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/68.jpg)
● 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
![Page 69: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/69.jpg)
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.
![Page 70: Quantum Algorithms Lecture #3ias.huji.ac.il/sites/default/files/quantum_algorithms_lecture3.pdf · Knot Invariants Lacking an algorithm for UNKNOT, one can make partial progress with](https://reader035.fdocuments.net/reader035/viewer/2022081222/5f7a9934c313b20b6c5b3317/html5/thumbnails/70.jpg)
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