Transversal logical gates

Transcription

1 Transversal logical gates SPT phases gapped boundaries eni Yoshida (Perimeter Institute) *SPT = symmetry protected topological

2 Transversal logical gates SPT phases gapped boundaries *SPT = symmetry protected topological

3 Transversal Logical Gates at QIP ariv: Sergey ravyi, Robert Koenig ariv: dam Paetznick, en Reichardt ariv: Hector ombin ariv: Fernando Pastawski, eni Yoshida ariv: lex Kubica, eni Yoshida, Fernando Pastawski ariv: eni Yoshida ariv: eni Yoshida ariv: eni Yoshida ariv: Sergey ravyi, ndrew ross QIP13 QIP14 QIP15 QIP15 Later This talk Merged Later This talk remark ariv: will not be covered in this talk.

4 Why transversal gates? The problem(s) Given a quantum error-correcting code, how do we find transversal logical gates? How do we design a quantum error-correcting code with useful transversal logical gates? input psi> > > > > > > Ideally, by transversal implementation U U U U U U U encoding circuit

5 ravyi-koenig theorem (212)

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7 recalling construction of the quantum double model in d-spatial dimensions [46, 47]. onsider a 1 g v. with the orthogonal basis { gi : g 2 G} (1) is associated to ice with directed edges. v =Hilbert space G g2g G is a finite group. The entire Hilbert space is denoted by H1. Operators g v and p are defined d p are projectors and pairwise commute. The Hamiltonian of the d-dimensional quantum double 1 where p is a projector onto a subspace with no flux on a plaquette p. Define Quantum double model in d dimensions 1 g Quantum double v =v is apcertain v. family of topological codes. HG = model (2) G v p (1) g2g Given a d-dimensional directed graph and a finite group G, one tions run over all vertices v and plaquettes p. The ground state gs i satisfies p are projectors and pairwise commute. The Hamiltonian of the d-dimensional quantum double can define the quantum double model. v gs i = p gs i = gs i (3) arbitrary closed loop, one can also operator HGdefine = a projection v p ( ) onto fluxless subspaces. We G -dimensional v p gs i = gs i for any contractible loop. 14.Yu. Kitaev / nnals of Physics 33 (23) 2 3 tions run over all vertices v and plaquettes p. The ground state v gs i = p gs i = gs i (2) spin gs i satisfies (3) arbitrary closed loop, one can also define a projection operator ( ) onto fluxless subspaces. We gs i = gs i for any contractible loop. Fig. 6. Generic lattice and the orientation rules for the operators L of operators g v doesg not depend on p,gwe retain this parameter to emphasize t 1 and p. Here I represents an identity element in G, L+pÞ hiand = ghi and 11L These hi = operators hg i. generate an algebra D ¼ g ðs; leads h ðs; pþ. to *One can also add twists, which the Dijkgraaf-Witten model.

8 Main result systematic framework for constructing logical gates We will consider d-cocycle functions over G by studying the group cohomology Hd(G,U(1)).! n (g 1,...,g n ) Using d-cocycle functions, we can provide a recipe of constructing a fault-tolerant logical gate for the d-dimensional quantum double model. If the cocycle function has a non-trivial sequence of slant Y products, then the logical gate is a non-trivial d-th level gate. *Slant product : a map from n-cocycle to n-1 cocycle! n! ig1! (g 1) n! ig2! (g 1,g 2 ) n!!! (g 1,g 2,...,g n 1 ) n! ig n n-cocyle (n-1)-cocyle -cocyle! (g 1,g 2,...,g n 1,g n ) n U(1) phase ariv: Y

9 Transversal logical gates SPT phases gapped boundaries

10 lassification of gapped boundaries The Toric code has two types of boundaries (ravyi-kitaev 98) electric charge Which anyons can *condense into a boundary? rough boundary (i) rough boundary; electric charge (ii) smooth boundary; magnetic flux magnetic flux smooth boundary * can create and annihilate an anyon without involving others.

11 Lagrangian subgroup [Levin 13] ondensing anyons are characterized by Lagrangian subgroup. topological phase : set of anyons boundary Lagrangian subgroup (1) ll the anyons in braid trivially with each other. (2) Maximal subset. (lmost) complete classification of 2dim gapped boundaries ond-mat High-Energy Quant-Info Math ais-slingerland 9, Kapustin-Saulina 11, Levin-Gu 12, Levin 13, Wang-Wen 12, arkeshli-jian-qi 13, Lan-Wang-Wen 15 ravyi-kitaev 98, ombin-martindelgado 8, ombin 1, eigi-shor-whalen 11 Kitaev-Kong 12, Kong 13, Fuchs-Shweigert-Valentino 14

12 Gapped boundary and logical gate Topological color code Hadamard logical gate is transversal. ariv: Y Fault-tolerant logical gates and gapped domain walls are closely related.

13 Gapped boundary and logical gate Topological color code Hadamard logical gate is transversal. pply Hadamard only on the right hand side ariv: Y Fault-tolerant logical gates and gapped domain walls are closely related.

14 Gapped boundary and logical gate Fault-tolerant logical gates and gapped domain walls are closely related. gapped domain wall Hadamard logical gate is transversal. e e ec m m domain wall m m mc e e m e Topological color code pply Hadamard only on the right hand side ariv: Y

15 Domain walls vs logical gates Fact Given a fault-tolerant logical gate, one can construct a transparent domain wall in topological quantum code. onjecture There is a one-to-one correspondence between transparent domain walls and fault-tolerant logical gates in topological quantum field theory (TQFT). For 2*2, there are 72 different domain walls. ll of them have corresponding logical operations.

16 nother result We construct a gapped boundary / gapped domain wall in the d- dimensional quantum double model by using d-cocycle functions. In d>2, we can construct a gapped boundary where none of anyonic excitations can condense. (No electric charge/magnetic flux can condense) nyons can condense into a boundary only if they are accompanied by superpositions of anyonic excitations. Lagrangian subgroup needs to be modified.

17 Transversal logical gates SPT phases gapped boundaries

18 Topological color code (ombin) The Hamiltonian is given by H = P S () P P S () P defined on a three-colorable lattice string operators anyons in the color code e e e m m m Pauli along Pauli along Pauli along Pauli along Pauli along Pauli along (equivalent to two copies of the toric code, Y21)

19 Membrane operators in the color code (1) Hadamard operator H :!!! (2) R2 Phase operator 2 3 R 2 = e i /2 H!! R 2 :! Y Y!! omment: Rm operators are transversal in m-dimensional color code R m := diag(1, exp(i /2 m 1 )).

20 String-like excitations? String logical operators Membrane logical operators Point-like anyonic excitations String-like anyonic excitations(?) R2 operators superposition of electric charges (?)

21 nswer Loop-like excitations in the 2dim color code are characterized by 1dim 2*2 SPT phase. psi > = + e i 1 + e i 2 R2 operators applied +

22 nswer Loop-like excitations in the 2dim color code are characterized by 1dim 2*2 SPT phase. 1 1 psi > = + e i 1 + e i R2 operators applied + can be viewed as a one-dimensional wavefunction

23 Why SPT phases? Origin of symmetries Parity constraints of electric charges 2dim color code = 2 copies of the toric code Electric charges from copy and copy get entangled to form a looplike object. Origin of non-triviality Non-triviality of the gapped domain wall. SPT excitations

24 Toward classification of logical gates Fault-tolerant logical gates Possible excitations Group cohomology Hadamard gate R2 gate superpositions of anyons Pauli- Pauli- anyons

25 Key idea: sweeping SPT excitations Sweep the domain wall over the entire system. SPT phases are characterized by cocycle functions. Logical actions are characterized by cocycle functions. quantum double model SPT excitations quantum double model gapped domain wall

26 Topological color code? * The d-dimensional topological color code has a transversal Rd phase gate which belongs to the d-th level (outside of d-1 th level). (ombin7) * d-dimensional color code is equivalent to d copies of the d-dimensional toric code. (Kubica-Y-Pastawski 15) i.e. the d-dimensional quantum double model with G =( 2 ) d i * There is a non-trivial d-cocycle:! d (g 1,...,g d )=( 1) g(1) 1...g(d) d w * The corresponding gate is the d-qubit control- gate.

27 Overview of the results Transversal logical gates for d-dim quantum double model SPT phases group cohomology (d-cocycle) gapped boundaries beyond Lagrangian subgroup

28 Domain wall in three-dimensions magnetic flux becomes a composite of magnetic flux and superposition of electric charges (3dim color code) R3 operators only on the right hand side

29 Three-loop braiding statistics The three-dimensional color code exhibits non-trivial braiding statistics. flux SPT flux The statistical angle can be computed by taking slant products of cocycle functions.

30 perform entangled measurements on four-body (78) trast, Renyi-2 of entropies an the isomorphic state can trast, Renyi-2 of an isomorphic stateb measurements of spinsentropies by reporting experiments mult measurements of spins by the reporting the experimm measurements of spins by reporting experiments gapped walls willdomain work only for in a small system size due to the cost of will a work only system for a small system size due tocos th will work only for small size due to the e, SPT excitations an indirect way need of measuring correlation func an indirectout-of-time way of measuring out-of-time correl braiding statistics studied by a out-of-time group commutator Two-particle an indirect waycanofbemeasuring correlation 3 (1, g, g1, g2 ) 2 U (1) 3(1, g, g1, g2) 2 U (1) 3 (1, g, g1, g2 ) 2 U (1) K(U, U ) = U U U U Multi-excitation braiding transformations implements the three-loop braiding: Three-loop braiding statistics can be studied by a sequential group commutator K(K(U, U ), U ) gs i = ei (,, ) gs i, where K(K(U, U ), U ) = (U U U U ) U (U U U U )U. s such, the three-loop braiding statistics corresponds to the vacuum expectation va commutator.

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32 Overview of the results Transversal logical gates for d-dim quantum double model SPT phases group cohomology (d-cocycle) gapped boundaries beyond Lagrangian subgroup ariv: ariv:

33 Overview of the results Transversal logical gates for d-dim quantum double model SPT phases group cohomology (d-cocycle) gapped boundaries beyond Lagrangian subgroup SPT phases beyond group cohomology ariv: