Universal Quantum Computation with Gapped Boundaries

Iris Cong, Meng Cheng, Zhenghan Wang

[email protected]

Department of Physics Harvard University, Cambridge, MA

May 4th, 2019

Refs.: C, Cheng, Wang, Comm. Math. Phys. (2017) 355: 645. C, Cheng, Wang, Phys. Rev. Lett. 119, 170504 (2017).

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Major challenge: local decoherence of qubits

Introduction: Quantum Computation

Quantum computing: Feynman (1982): Use a quantum-mechanical computer to simulate quantum physics (classically intractable) Encode information in qubits, gates = unitary operations Applications: Factoring/breaking RSA (Shor, 1994), quantum machine learning, ...

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Introduction: Quantum Computation

Quantum computing: Feynman (1982): Use a quantum-mechanical computer to simulate quantum physics (classically intractable) Encode information in qubits, gates = unitary operations Applications: Factoring/breaking RSA (Shor, 1994), quantum machine learning, ... Major challenge: local decoherence of qubits

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Introduction: Topological Quantum Computation

Topological quantum computing (TQC) (Kitaev, 1997; Freedman et al., 2003): Encode information in topological degrees of freedom Perform topologically protected operations

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Qubit encoding: degeneracy arising from the fusion rules e.g. Toric code: ei mj ek ml = e(i+k)(mod 2)m(j+l)(mod 2) (i, j =0, 1) e.g. Ising: =1⌦ , =1 e.g. Fibonacci:⌦ ⌧ ⌧ =1 ⌧ ⌦

Topological Quantum Computation

Traditional realization of TQC: anyons in topological phases of matter (e.g. Fractional Quantum Hall - FQH) Elementary quasiparticles in 2 dimensions s.t. = ei | 1 2i | 2 1i

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Topological Quantum Computation

Traditional realization of TQC: anyons in topological phases of matter (e.g. Fractional Quantum Hall - FQH) Elementary quasiparticles in 2 dimensions s.t. = ei | 1 2i | 2 1i Qubit encoding: degeneracy arising from the fusion rules e.g. Toric code: ei mj ek ml = e(i+k)(mod 2)m(j+l)(mod 2) (i, j =0, 1) e.g. Ising: =1⌦ , =1 e.g. Fibonacci:⌦ ⌧ ⌧ =1 ⌧ ⌦

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Topological Quantum Computation

Topologically protected operations: Braiding of anyons Move one anyon around another pick up phase (due to Aharonov-Bohm) !

Figures: (1) Z. Wang, Topological Quantum Computation. (2) C. Nayak et al., Non-abelian anyons and topological quantum computation.

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Answer: Yes! We consider boundaries of the topological phase gapped boundaries ! We’ll even get a universal gate set from gapped boundaries of an abelian phase (specifically, bilayer ⌫ =1/3 FQH)

These are dicult to realize, existence is still uncertain Question: Given a top. phase that supports only abelian anyons, is it possible “engineer” other non-abelian objects?

Problem

Problem: Abelian anyons have no degeneracy, so no computation power need non-abelian anyons for TQC (e.g. ⌫ =5/2, 12/5) !

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Answer: Yes! We consider boundaries of the topological phase gapped boundaries ! We’ll even get a universal gate set from gapped boundaries of an abelian phase (specifically, bilayer ⌫ =1/3 FQH)

Question: Given a top. phase that supports only abelian anyons, is it possible “engineer” other non-abelian objects?

Problem

Problem: Abelian anyons have no degeneracy, so no computation power need non-abelian anyons for TQC (e.g. ⌫ =5/2, 12/5) ! These are dicult to realize, existence is still uncertain

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Answer: Yes! We consider boundaries of the topological phase gapped boundaries ! We’ll even get a universal gate set from gapped boundaries of an abelian phase (specifically, bilayer ⌫ =1/3 FQH)

Problem

Problem: Abelian anyons have no degeneracy, so no computation power need non-abelian anyons for TQC (e.g. ⌫ =5/2, 12/5) ! These are dicult to realize, existence is still uncertain Question: Given a top. phase that supports only abelian anyons, is it possible “engineer” other non-abelian objects?

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Problem

Problem: Abelian anyons have no degeneracy, so no computation power need non-abelian anyons for TQC (e.g. ⌫ =5/2, 12/5) ! These are dicult to realize, existence is still uncertain Question: Given a top. phase that supports only abelian anyons, is it possible “engineer” other non-abelian objects? Answer: Yes! We consider boundaries of the topological phase gapped boundaries ! We’ll even get a universal gate set from gapped boundaries of an abelian phase (specifically, bilayer ⌫ =1/3 FQH)

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Overview

Framework of TQC Introduce gapped boundaries and their framework Gapped boundaries for TQC Universal TQC with gapped boundaries in bilayer ⌫ =1/3 FQH Summary and Outlook

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Overview

Framework of TQC Introduce gapped boundaries and their framework Gapped boundaries for TQC Universal TQC with gapped boundaries in bilayer ⌫ =1/3 FQH Summary and Outlook

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 c For each pair of anyon types, a set of fusion rules: a b = c Nabc. ⌦ c 1 The fusion space of a b is a vector space Vab with basis Vab. ⌦ b The fusion space of a1 a2 ... an to b is a vector space Va1a2...an with basis given by anyon⌦ labels⌦ in⌦ intermediate segments, e.g.

Framework of TQC

Formally, an anyon model consists of the following data: B Set of anyon types/labels: a, b, c... , one of which should represent the vacuum{ 1 } Each anyon type has a topological twist ✓ U(1): i 2

1 More general cases exist, but are not used in this talk. Figures: Z. Wang, Topological Quantum Computation (2010) Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Framework of TQC

Formally, an anyon model consists of the following data: B Set of anyon types/labels: a, b, c... , one of which should represent the vacuum{ 1 } Each anyon type has a topological twist ✓ U(1): i 2 c For each pair of anyon types, a set of fusion rules: a b = c Nabc. ⌦ c 1 The fusion space of a b is a vector space Vab with basis Vab. ⌦ b The fusion space of a1 a2 ... an to b is a vector space Va1a2...an with basis given by anyon⌦ labels⌦ in⌦ intermediate segments, e.g.

1 More general cases exist, but are not used in this talk. Figures: Z. Wang, Topological Quantum Computation (2010) Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Framework of TQC

Formally, an anyon model consists of the following data: [Cont’d] B Associativity: for each (a, b, c, d), a set of (unitary) linear transformations F abc : V abc V abc satisfying { d;ef d ! d } “pentagons”

Figures: Z. Wang, Topological Quantum Computation (2010) Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Mathematically, this is captured by a(unitary)modular tensor category (UMTC)

Framework of TQC

Formally, an anyon model consists of the following data: [Cont’d] B (Non-degenerate) braiding: For each (a, b, c)s.t.Nc = 0, a set a,b 6 of phases1 Rc U(1) compatible ab 2 with associativity (“hexagons”):

1 More general cases exist, but are not used in this talk. Figures: Z. Wang, Topological Quantum Computation (2010) Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Framework of TQC

Formally, an anyon model consists of the following data: [Cont’d] B (Non-degenerate) braiding: For each (a, b, c)s.t.Nc = 0, a set a,b 6 of phases1 Rc U(1) compatible ab 2 with associativity (“hexagons”):

Mathematically, this is captured by a(unitary)modular tensor category (UMTC)

1 More general cases exist, but are not used in this talk. Figures: Z. Wang, Topological Quantum Computation (2010) Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Framework of TQC

In this framework, a 2D topological phase of matter is an equivalence class of gapped Hamiltonians = H whose H { } low-energy excitations form the same anyon model B Examples (on the lattice) include Kitaev’s quantum double models, Levin-Wen string-net models, ...

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Framework of TQC: Example

Physical system: Bilayer FQH system, 1/3 Laughlin state of opposite chirality in each layer

Equivalent to Z3 toric code (Kitaev, 2003) Anyon types: eamb, a, b =0, 1, 2 Twist: ✓(eamb)=!ab where ! = e2⇡i/3 Fusion rules: eamb ec md e(a+c)(mod 3)m(b+d)(mod 3) ⌦ ! F symbols all trivial (0 or 1) 2⇡ibc/3 Reamb,ec md = e

UMTC: SU(3)1 SU(3)1 = D(Z3)= (Rep(Z3)) = (Vec 3 ) ⇥ ⇠ Z Z Z

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Overview

Framework of TQC Introduce gapped boundaries and their framework Gapped boundaries for TQC Universal gate set with gapped boundaries in bilayer ⌫ =1/3 FQH Summary and Outlook

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 In the anyon model: Collection of bulk bosonic (✓ = 1) anyons which condense to vacuum on the boundary (think Bose condensation) All other bulk anyons condense to confined “boundary excitations” ↵, , ... Mathematically, Lagrangian algebra A 2 B

Gapped Boundaries: Framework

A gapped boundary is an equivalence class of gapped local (commuting) extensions of H to the boundary 2 H

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Gapped Boundaries: Framework

A gapped boundary is an equivalence class of gapped local (commuting) extensions of H to the boundary 2 H In the anyon model: Collection of bulk bosonic (✓ = 1) anyons which condense to vacuum on the boundary (think Bose condensation) All other bulk anyons condense to confined “boundary excitations” ↵, , ... Mathematically, Lagrangian algebra A 2 B

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 (More generally, we also define these with boundary excitations, but that is unnecessary for this talk.) M symbols must be compatible with F symbols (“mixed pentagons”):

Gapped Boundaries: Framework

More rigorously, gapped boundaries come with M symbols (like F symbols for the bulk):

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 M symbols must be compatible with F symbols (“mixed pentagons”):

Gapped Boundaries: Framework

More rigorously, gapped boundaries come with M symbols (like F symbols for the bulk):

(More generally, we also define these with boundary excitations, but that is unnecessary for this talk.)

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Gapped Boundaries: Framework

More rigorously, gapped boundaries come with M symbols (like F symbols for the bulk):

(More generally, we also define these with boundary excitations, but that is unnecessary for this talk.) M symbols must be compatible with F symbols (“mixed pentagons”):

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Two gapped boundary types: 2 Electric charge condensate: 1 =1 e e Magnetic flux condensate: A =1 m m2 A2

Gapped Boundaries: Example

Physical system: Bilayer ⌫ =1/3 FQH, equiv. to Z3 toric code Anyon types: eamb, a, b =0, 1, 2

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Gapped Boundaries: Example

Physical system: Bilayer ⌫ =1/3 FQH, equiv. to Z3 toric code Anyon types: eamb, a, b =0, 1, 2 Two gapped boundary types: 2 Electric charge condensate: 1 =1 e e Magnetic flux condensate: A =1 m m2 A2

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 M symbols for this theory are all 0 or 1

Gapped Boundaries: Example

Physical system: Bilayer ⌫ =1/3 FQH, equiv. Z3 toric code We will work mainly with =1 e e2: A1 Algebraically, 1, 2 are equivalent by electric-magnetic duality Easier to workA withA charge condensate - read-out can be done by measuring electric charge (Barkeshli, 2016) It is interesting to consider both 1 and 2 at the same time - we do this in a separate paper3, will brieflyA mentionA in our Outlook

3C, Cheng, Wang, Phys. Rev. B 96, 195129 (2017) Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Gapped Boundaries: Example

Physical system: Bilayer ⌫ =1/3 FQH, equiv. Z3 toric code We will work mainly with =1 e e2: A1 Algebraically, 1, 2 are equivalent by electric-magnetic duality Easier to workA withA charge condensate - read-out can be done by measuring electric charge (Barkeshli, 2016) It is interesting to consider both 1 and 2 at the same time - we do this in a separate paper3, will brieflyA mentionA in our Outlook M symbols for this theory are all 0 or 1

3C, Cheng, Wang, Phys. Rev. B 96, 195129 (2017) Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Overview

Framework of TQC Introduce gapped boundaries and their framework Gapped boundaries for TQC Universal gate set with gapped boundaries in bilayer ⌫ =1/3 FQH Summary and Outlook

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 For qudit encoding: use n =2

Gapped Boundaries for TQC

Qudit encoding: Gapped boundaries give rise to a natural ground state degeneracy: n gapped boundaries on a plane, with total charge vacuum

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Gapped Boundaries for TQC

Qudit encoding: Gapped boundaries give rise to a natural ground state degeneracy: n gapped boundaries on a plane, with total charge vacuum For qudit encoding: use n =2

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Gapped Boundaries for TQC: Example

For our bilayer ⌫ =1/3 FQH system, we have:

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Gapped Boundaries for TQC

Topologically protected operations: Tunnel-a operations Loop-a operations Braiding gapped boundaries

Topological charge measurement⇤

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Tunnel-a Operations

Starting from state b ,tunnelana anyon from to : | i A1 A2

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Tunnel-a Operations

Compute using M-symbols:

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Tunnel-a Operations

Result:

ab ab dadb W () = M ( )[M ]†( ) (1) a c A1 c A2 d c r c X

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Loop-a Operations

Starting from state b , loop an a anyon around one of the boundaries: | i

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Loop-a Operations

Similar computation methods lead to the formula:

where sab =˜sab/db is given by the modular S matrix of the theory:

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Braiding Gapped Boundaries

Braid gapped boundaries around each other:

(Mathematically, this gives a representation of the (spherical) 2n-strand pure braid group.)

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Braiding Gapped Boundaries

Simplify with (bulk) R and F moves to get:

2 a2a1b1 b1a1 a b a2a1b1 1 = F R R 1 1 (F ) (2) 2 b2;c0c c0 c b2 c00c0 Xc,c0

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Gapped Boundaries for TQC: Example

For our bilayer ⌫ =1/3 FQH case: ( = =1 e e2) A1 A2 Tunnel an e anyon from to : W () b = a b A1 A2 a | i | ⌦ i W ()=x ,wherex i = (i + 1)(mod 3) ! e 3 3 | i | i Loop an m anyon around : W (↵ ) ej = !j ej A2 m 2 | i | i W (↵ )=z ! m 2 3 Braid gapped boundaries: get z ^ 3

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Topological charge projection (TCP) (Barkeshli and Freedman, 2016): Doubled theories: Wilson line lifts to a loop measure topological charge through the loop !

Resulting projection operator: ( ()=W ( )orW (↵ )) Ox xx i x i (a) P = S S ⇤ (). (3) 0a xaOx x X2C

Topological Charge Projection/Measurement

Motivation: Property F conjecture (Naidu and Rowell, 2011): Braidings alone cannot be universal for TQC for most physically plausible systems

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Topological Charge Projection/Measurement

Motivation: Property F conjecture (Naidu and Rowell, 2011): Braidings alone cannot be universal for TQC for most physically plausible systems Topological charge projection (TCP) (Barkeshli and Freedman, 2016): Doubled theories: Wilson line lifts to a loop measure topological charge through the loop !

Resulting projection operator: ( ()=W ( )orW (↵ )) Ox xx i x i (a) P = S S ⇤ (). (3) 0a xaOx x X2C

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 General topological charge measurements (TCMs): (a) Projection operators P = x S0aSxa⇤ x () topological charge 2C O ! (a)1 measurements perform the complementP of P Not always physical, but special cases are symmetry protected –we examine this

Topological Charge Projection/Measurement

Topological charge projection (TCP): [Cont’d] Given an anyon theory ,its , matrices C S T

= , = diag(✓i ) S ( ) T

give mapping class group representations V (Y )forsurfacesY . Barkeshli and Freedman showed that topologicalC charge projections generate all matrices in V (Y ) C

1 More general cases exist, but are not used in this talk. Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Topological Charge Projection/Measurement

Topological charge projection (TCP): [Cont’d] Given an anyon theory ,its , matrices C S T

= , = diag(✓i ) S ( ) T

give mapping class group representations V (Y )forsurfacesY . Barkeshli and Freedman showed that topologicalC charge projections generate all matrices in V (Y ) C General topological charge measurements (TCMs): (a) Projection operators P = x S0aSxa⇤ x () topological charge 2C O ! (a)1 measurements perform the complementP of P Not always physical, but special cases are symmetry protected –we examine this

1 More general cases exist, but are not used in this talk. Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Overview

Framework of TQC Introduce gapped boundaries and their framework Gapped boundaries for TQC Universal gate set with gapped boundaries in bilayer ⌫ =1/3 FQH Summary and Outlook

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Universal Gate Set

Universal (metaplectic) gate set for the qutrit model (Cui and Wang, 2015):

1 The single-qutrit Hadamard gate H3,definedas H j = 1 2 !ij i , j =0, 1, 2, ! = e2⇡i/3 3| i p3 i=0 | i 2 The two-qutritP entangling gate SUM3,definedas SUM i j = i (i + j)mod3, i, j =0, 1, 2. 3| i| i | i| i 3 The single-qutrit generalized phase gate Q3 = diag(1, 1, !). 4 Any nontrivial single-qutrit classical (i.e. Cli↵ord) gate not equal to 2 H3 . 5 A projection M of a state in the qutrit space C3 to Span 0 and {| i} its orthogonal complement Span 1 , 2 ,sothattheresultingstate {| i | i} is coherent if projected into Span 1 , 2 . {| i | i}

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 2 z can be implemented by gapped boundary braiding. Conjugate ^ 3 second qutrit by H3 to get SUM3. 3 TCP can implement diag(1, !, !) (Dehn twist of SU(3)1). Follow by x 3 for Q3. 4 Any nontrivial single-qutrit classical (i.e. Cli↵ord) gate not equal to 2 H3 -wehaveaPauli-X from tunneling e. 5 Projective measurement - we use the TCM which is the complement of 111 1 P(1) = 111. (4) 3 2 3 111 (1) 4 5 Conjugating 1 P with the Hadamard gives the result.

Universal Gate Set - Bilayer ⌫ =1/3FQH

Universal (metaplectic) gate set for the qutrit model (Cui and Wang, 2015): 1 H = for = SU(3) (single layer ⌫ =1/3 FQH), so it can be 3 S C 1 implemented by TCP.

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 3 TCP can implement diag(1, !, !) (Dehn twist of SU(3)1). Follow by x 3 for Q3. 4 Any nontrivial single-qutrit classical (i.e. Cli↵ord) gate not equal to 2 H3 -wehaveaPauli-X from tunneling e. 5 Projective measurement - we use the TCM which is the complement of 111 1 P(1) = 111. (4) 3 2 3 111 (1) 4 5 Conjugating 1 P with the Hadamard gives the result.

Universal Gate Set - Bilayer ⌫ =1/3FQH

Universal (metaplectic) gate set for the qutrit model (Cui and Wang, 2015): 1 H = for = SU(3) (single layer ⌫ =1/3 FQH), so it can be 3 S C 1 implemented by TCP. 2 z can be implemented by gapped boundary braiding. Conjugate ^ 3 second qutrit by H3 to get SUM3.

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 4 Any nontrivial single-qutrit classical (i.e. Cli↵ord) gate not equal to 2 H3 -wehaveaPauli-X from tunneling e. 5 Projective measurement - we use the TCM which is the complement of 111 1 P(1) = 111. (4) 3 2 3 111 (1) 4 5 Conjugating 1 P with the Hadamard gives the result.

Universal Gate Set - Bilayer ⌫ =1/3FQH

Universal (metaplectic) gate set for the qutrit model (Cui and Wang, 2015): 1 H = for = SU(3) (single layer ⌫ =1/3 FQH), so it can be 3 S C 1 implemented by TCP. 2 z can be implemented by gapped boundary braiding. Conjugate ^ 3 second qutrit by H3 to get SUM3. 3 TCP can implement diag(1, !, !) (Dehn twist of SU(3)1). Follow by x 3 for Q3.

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 5 Projective measurement - we use the TCM which is the complement of 111 1 P(1) = 111. (4) 3 2 3 111 (1) 4 5 Conjugating 1 P with the Hadamard gives the result.

Universal Gate Set - Bilayer ⌫ =1/3FQH

Universal (metaplectic) gate set for the qutrit model (Cui and Wang, 2015): 1 H = for = SU(3) (single layer ⌫ =1/3 FQH), so it can be 3 S C 1 implemented by TCP. 2 z can be implemented by gapped boundary braiding. Conjugate ^ 3 second qutrit by H3 to get SUM3. 3 TCP can implement diag(1, !, !) (Dehn twist of SU(3)1). Follow by x 3 for Q3. 4 Any nontrivial single-qutrit classical (i.e. Cli↵ord) gate not equal to 2 H3 -wehaveaPauli-X from tunneling e.

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Universal Gate Set - Bilayer ⌫ =1/3FQH

Universal (metaplectic) gate set for the qutrit model (Cui and Wang, 2015): 1 H = for = SU(3) (single layer ⌫ =1/3 FQH), so it can be 3 S C 1 implemented by TCP. 2 z can be implemented by gapped boundary braiding. Conjugate ^ 3 second qutrit by H3 to get SUM3. 3 TCP can implement diag(1, !, !) (Dehn twist of SU(3)1). Follow by x 3 for Q3. 4 Any nontrivial single-qutrit classical (i.e. Cli↵ord) gate not equal to 2 H3 -wehaveaPauli-X from tunneling e. 5 Projective measurement - we use the TCM which is the complement of 111 1 P(1) = 111. (4) 3 2 3 111 (1) 4 5 Conjugating 1 P with the Hadamard gives the result. Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Implementing M ground state of H is doubly degenerate for $ 0 e , e2 t is real (beyond topological protection) | i | i$ Physically, could realize in fractional quantum spin Hall state – quantum spin Hall + time-reversal symmetry (exchange two layers) Topologically equiv. to ⌫ =1/3 Laughlin, e = bound state of spin up/down quasiholes e is time-reversal invariant tunneling amplitude of e must be real symmetry-protected TCM! !

Universal Gate Set - Bilayer ⌫ =1/3FQH

To get the projective measurement, we introduce a symmetry-protected topological charge measurement: Want to tune system s.t. quasiparticle tunneling along is enhanced implement H = tW (e)+h.c. ! 0 t = (complex) tunneling amplitude, W (e)=tunnel-e operator

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Physically, could realize in fractional quantum spin Hall state – quantum spin Hall + time-reversal symmetry (exchange two layers) Topologically equiv. to ⌫ =1/3 Laughlin, e = bound state of spin up/down quasiholes e is time-reversal invariant tunneling amplitude of e must be real symmetry-protected TCM! !

Universal Gate Set - Bilayer ⌫ =1/3FQH

To get the projective measurement, we introduce a symmetry-protected topological charge measurement: Want to tune system s.t. quasiparticle tunneling along is enhanced implement H = tW (e)+h.c. ! 0 t = (complex) tunneling amplitude, W (e)=tunnel-e operator Implementing M ground state of H is doubly degenerate for $ 0 e , e2 t is real (beyond topological protection) | i | i$

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Universal Gate Set - Bilayer ⌫ =1/3FQH

To get the projective measurement, we introduce a symmetry-protected topological charge measurement: Want to tune system s.t. quasiparticle tunneling along is enhanced implement H = tW (e)+h.c. ! 0 t = (complex) tunneling amplitude, W (e)=tunnel-e operator Implementing M ground state of H is doubly degenerate for $ 0 e , e2 t is real (beyond topological protection) | i | i$ Physically, could realize in fractional quantum spin Hall state – quantum spin Hall + time-reversal symmetry (exchange two layers) Topologically equiv. to ⌫ =1/3 Laughlin, e = bound state of spin up/down quasiholes e is time-reversal invariant tunneling amplitude of e must be real symmetry-protected TCM! !

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Overview

Framework of TQC Introduce gapped boundaries and their framework Gapped boundaries for TQC Universal gate set with gapped boundaries in bilayer ⌫ =1/3 FQH Summary and Outlook

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Summary

Decoherence is a major challenge to quantum computing ! topological quantum computing (TQC) TQC with anyons requires non-abelian topological phases (dicult to implement) engineer non-abelian objects (e.g. gapped ! boundaries) from abelian phases We can get a universal quantum computing gate set from a purely abelian theory (bilayer ⌫ =1/3 FQH), which is trivial for anyonic TQC Topologically protected qudit encoding and Cli↵ord gates Symmetry-protected implementation for non-Cli↵ord projection

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Outlook

Practical implementation of the symmetry-protected TCM More thorough study of symmetry-protected quantum computation Amount of protection o↵ered and computation power Other routes to engineer non-abelian objects Boundary defects/parafermion zero modes from gapped boundaries of ⌫ =1/3 FQH (Lindner et al., 2014) Genons and symmetry defects (Barkeshli et al., 2014; C, Cheng, Wang, 2017; Delaney and Wang, 2018) How would these look when combined with gapped boundaries?

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Acknowledgments

Special thanks to Cesar Galindo, Shawn Cui, Maissam Barkeshli for answering many questions Many thanks to Prof. Freedman and everyone at Station Q for a great summer None of this would have been possible without the guidance and dedication of Prof. Zhenghan Wang

Iris Cong, Harvard TQC with Gapped Boundaries NCGOA, May 2019 Quantum Convolutional Neural Networks

Iris Cong Soonwon Choi Mikhail D. Lukin arXiv:1810.03787 Why quantum machine learning? Machine learning: interpret and process large amounts of data

Unmanned Vehicle Genomics Image recognition

= Cat

= Dog

Quantum physics: many-body interactionsà extremely large complexity Near-term quantum computers/ Quantum chemistry (a) Quantum phases of matter quantum simulators (b) X X X FC

Paramagnetic d !… Z X Z X Z … !⇥ ! Z X Z X Z ! P

SPT ! ! X X X X X X X ! ! … … C Antiferromagnetic C H. Bernien et al, Nature (2017)

Machine learning(c) Quantum many-body physics(d) Exciting! Quantum Machine Learning

• Using a quantum computer to perform machine learning tasks

• Many open questions:

• Why/how does quantum machine learning work?

• Concrete circuit models suitable for near-term implementation?

• Relationship to quantum many-body physics?

• Relationship with quantum information theory? Goal of our work (a) 0 l ⇢ | i | 1i 1 1 1 UU UU UU UU 1 1 2 2 1 1 2 2 NN Main Contributions 0 l ⇢ 0 ⇢ | i | i | i | li 0 l ⇢ 1 1 1 1 | i | 1i 1 1 11 1 0 Ul U ⇢ U U U1U1UUU2U2 UUU U1U1UU202U1 U 2l ⇢ 1 1 2 2 1 1 N N2 21 2 NN N | i | i 0 l ⇢ | i | i 0 ⇢ | i | i Concrete + efficient circuit model for Theoretical Explanation: | i | li 0 l ⇢ quantum classification problems Application: Quantum Application:| i | 1i1 1 Optimizing1 RG Flow, MERA, UU UU UU UU 1 1 2 2 1 1 2 2 NN Phase Recognition Quantum0 l Error⇢ Correction • Good connection to existing ML techniques Quantum Error Correction | i | i Encoding Decoding

/J 0.4 0.8 1.2 0 10-1 1.5 (b) -1.2 Cluster Model: QCNN Phase Diagram (N = 135 Spins) -2 1 10 -0.8

0.5-0.4 10-3 0 /J /J 0 2 10-4 h 0.4

-0.5 Logical Error Rate -5 Fully 0.8 10 -1

Connected 1.2 10-6 -1.5 10-4 10-3 10-2 10-1 0 0.5 1 1.5 Input Error Rate Convolution Pooling -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 h1/J Review of (Classical) CNN • Structured neural network: multiple layers of image processing Example (simplified):

= Cat Dog

Convolution Pooling Fully connected

Convolution layers Pooling layers Fully connected

!",$ = &'(",$ + &*("+,$+ &,(",$+*+ &-("+*,$+* Quantum CNN Architecture

Same types of layers: 1. Convolution • Local unitaries, trans. inv., 1D, 2D, 3D … 2. Pooling • Reduce system size • Final unitary depends on meas. outcomes 3. Fully connected • Non-local measurement

Total number of parameters ~ O(log N) Fully Convolution Pooling Connected Application: Quantum Phase Recognition

Problem: Given quantum many-body system in (unknown) ground state |01⟩, does |01⟩ belong to a particular quantum phase 3?

Direct analog of image classification, but intrinsically quantum problem Claim: Quantum CNN is very efficient in quantum phase recognition

if |01⟩ ∈ 3 |01⟩ if |01⟩ ∉ 3 Fully Connected

Convolution Pooling (a) (b) X X X Example: 1D ZXZ Model (G = ℤ ×ℤ SPT) FC 6 6 Paramagnetic d !… Z X Z X Z … !⇥ ! Z X Z X Z ! P

• SPT phase: cannot be detected by local order parameterSPT ! ! X X X X X X X ! ! … … C • Antiferromagnetic C

(a) (b) X X X FC (c) (d)

Paramagnetic d !… Z X Z X Z … !⇥ ! Z X Z X Z ! P

SPT ! ! X X X X X X X ! ! … … C Antiferromagnetic C

Phase diagram(c) is obtained from iDMRG with bond dimension 150. (d) Input states are obtained from DMRG with system sizes 45, 135, bond dimension 130. Circuit is performed using matrix-product state update. Example: 1D ZXZ Model (G = ℤ6×ℤ6 SPT)

• S = 1 Haldane phase transition:

• Same phase, map spin-1 to pair of

spin-1/2 Sample Complexity

• Existing approaches to detect SPT: measure nonzero expectation value of string order parameters (long operator product)

• Problem: expectation value vanishes near phase boundary à many repetitions

• QCNN: much sharper à fewer repetitions

• Quantify with sample complexity: How many copies of the input state

are required to determine with 95% confidence that |01⟩ ∈ 3? (a) (b) X X X FC

Paramagnetic d !… Z X Z X Z … !⇥ ! Z X Z X Z ! P

SPT ! ! X X X X X X X ! ! Sample Complexity… … C Antiferromagnetic C • Comparison with existing approaches: h1 = 0.5 J: (c) (d) string order parameters

• SOP (red): independent of string length

• QCNN (blue): much better and

improves with depth up to finite size C P C P FC (a) j

(`) = (`) (` 1) i Cat Dog (` 1) (`) (` + 1) (` +( 1)` 1) (` + 1) (b) (a) (b) 1.6 " " " FC 0.9 1.2 0.8 × & 0.8 Paramagnetic 0.7 ! " ! " ! " " 0.6 ⇢ 0.4 in ! ! ! " 0.5 " " "

/J … … 2 0 P

h 0.4 -0.4 SPT 0.3 0.2 -0.8 " "" "" "" "" "" "" "" "" "" " Why does it work?0.1 … … -1.2 0 C Antiferromagnetic C -1.6 -0.1 0 0.4 0.8 1.2 1.6 h1/J (c) (c) (d)

= (MERA) MERA QCNN = QEC (QCNN)

QCNN ≈ MERA + QEC ≈ “RG flow” Training: Example • N = 15 spins (depth 1) for simulations Paramagnetic • Initialize all unitaries to random values SPT • Train along h = 0 (solvable) 2 Antiferromagnetic • Gradient descent:

• Observation: training on 1D, solvable set can still produce the correct 2D phase diagram

• Demonstrates how QCNN structure avoids overfitting Optimizing Quantum Error Correction

Problem: Given a realistic but unknown error model, find a resource- efficient, fault-tolerant quantum error correction code to protect against these errors.

QCNN structure resembles nested quantum error correction, and can be used to simultaneously optimize encoder and decoder ⇢phys e ⇢l ⇢l -1 ⇢phys ⇢phys error QCNN …

QCNN … … ⇢l ⇢l Encoder Decoder Optimizing Quantum Error Correction

Error models:

• Isotropic depolarization (px = py = pz)

• Anisotropic depolarization

• Anisotropic depolarization + correlated error (Xi Xi+1) (a) 0 l ⇢ | i | 1i 1 1 1 UU UU UU UU 1 1 2 2 1 1 2 2 NN 0 l ⇢ 0 ⇢ | i | i | i | li 0 l ⇢ 1 1 1 1 | i | 1i 1 1 11 1 0 Ul U ⇢ U U U1U1UUU2U2 UUU U1UU1UU202U1 U 2l ⇢ 1 1 2 2 1 1 N N2 21 2 NN N | i | i 0 l ⇢ | i | i 0 ⇢ | i | i | i | li 0 l ⇢ | i | 1i1 1 1 Optimizing Quantum ErrorU Correction1U1 U2U2 U1U1 UU2 2 0 ⇢ NN | i | li Circuit structure: ResultsEncoding for correlated error: Decoding

(a) 0 l ⇢ (b) | i | 1i 1 1 1 UU UU UU UU -2 1 1 2 2 1 1 2 2 NN 10 0 l ⇢ 0 ⇢ | i | i | i | li 0 l ⇢ 1 1 1 1 | i | 1i 1 1 11 1 0 Ul U ⇢ U U U1U1UUU2U2 UUU U1UU1UU202U1 U 2l ⇢ 1 1 2 2 1 1 N N2 21 2 NN N -4 | i | i 0 l ⇢ | i | i 10 0 ⇢ | i | i | i | li 0 l ⇢ | i | 1i1 1 1 Logical Error Rate U1U1 U2U2 U1U1 UU2 2 0 ⇢ NN -6 | i | li 10 10 -4 10 -3 10 -2 10 -1 Encoding Decoding Input Error Rate

(b) 10 -2

10 -4 Logical Logical Error Rate

10 -6 10 -4 10 -3 10 -2 10 -1 Input Error Rate Summary

(a) 0 l ⇢ | i | 1i 1 1 1 • Concrete circuit model for quantum classification UU UU U1U1 UU2 2 1 1 2 2 NN 0 l ⇢ 0 ⇢ | i | i | i | li • 0 l ⇢ Application to quantum phase recognition: 1 1 1 1 | i | 1i 1 1 11 1 0 Ul U ⇢ U U U1U1UUU2U2 UUU U1UU1UU202U1 U 2l ⇢ 1 1 2 2 1 1 N N2 21 2 NN N | i | i 0 l ⇢ | i | i • ℤ ×ℤ 0 l ⇢ | i | i 1D SPT phase ( 6 6) | i | i 0 l ⇢ | i | 1i1 1 1 • Theoretical explanation: QCNN ≈ MERA + QEC ≈ RG flow U1U1 U2U2 U1U1 UU2 2 0 ⇢ NN | i | li • Optimizing Quantum Error(a) Correction Encoding Decoding (b) X X X FC (b) Paramagnetic 10 -2 d !… Z X Z X Z … !⇥ ! Z X Z X Z ! P

10 -4 Fully SPT Connected ! ! X X X X X X X ! ! Logical Logical Error Rate … … C

-6 Convolution Pooling Antiferromagnetic 10 C 10 -4 10 -3 10 -2 10 -1 Input Error Rate (c) (d) Thanks!