high thresholds in two dimensions Robert Raussendorf, University of - PowerPoint PPT Presentation

Fault-tolerant quantum computation - high thresholds in two dimensions Robert Raussendorf, University of British Columbia QEC11, University of Southern California December 5, 2011

Outline • Motivation • Topological codes, Example 1: Surface codes • Twists, color, subsystems • Topological codes, Example 2: Subsystem color codes • Remarks

Introduction Optical Lattice Greiner Lab Superconducting qubit (Delft) Monroe ion-photon link Nist racetrack ion trap local global architecture

Fault-tolerant quantum computation • Fault-tolerance is the art of maintaining the quantum speedup in the presence of decoherence. Fault-tolerance theorem ∗ : If for a universal quantum computer the noise per elementary operation is below a constant non- zero error threshold ǫ then arbitrarily long quantum computa- tions can be performed efficiently with arbitrary accuracy. Remaining questions: • What is the value of the error threshold? • What is the operational cost of fault-tolerance? *: Aharonov & Ben-Or (1996), Kitaev (1997), Knill & Laflamme & Zurek (1998), Aliferis & Gottesman & Preskill (2005)

superconducting฀ qubits quantum฀dots optical฀lattices segmented฀ ion฀traps Our setting • 2D, nearest-neighbor translation-invariant interaction . • High fault-tolerance threshold • Moderate overhead scaling

Part I: Fault-tolerant quantum computation with the surface code

Main ingredients 1. Fault-tolerance from surface codes [Kitaev 97]: Translation-invariant and short-range interaction. 2. Topological quantum gates via time-dependent bound- ary conditions. qubit lattice, spacing a holes in the code plane a macroscopic scale a log(# gates) microscopic scale 2 microscopic view macroscopic view

1.1 The surface code One qubit located on every edge syndrome at endpoint X X plaquette Z X harmless Z ZZ stabilizer error X B p X X X X X X site stabilizer harmful X A s error Z = Z Z Z Z Z Z X Z Z Z X = X | ψ � = A s | ψ � = B p | ψ � , ∀| ψ � ∈ H C , ∀ s, p. (1) • Surface codes are stabilizer codes associated with 2D lattices. • Only the homology class of an error chain matters. A. Kitaev,quant-ph/9707021 (1997).

1.2 Topological error correction • Fault-tolerant data storage with planar code described by Random plaquette Z 2 -gauge model (RPGM) [1]. [1] Dennis et al. , quant-ph/0110143 (2001).

1.4 How to encode qubits • Storage capacity of the code depends upon the topology of the code surface.

1.5 Surface code on plane with holes XX site stabilizer not enforced X X primal hole Z plaquette stabilizer not enforced Z Z Z dual hole • There are two types of holes: primal and dual. • A pair of same-type holes constitutes a qubit.

฀hole primal dual hole dual hole rough฀boundary smooth฀boundary ฀hole primal 1.5 Surface code on plane with holes Surface code with boundary: primal qubit dual qubit X p Z d p Z p X d • X -chain cannot end in primal hole, can end in dual hole. • Z -chain can end in primal hole, cannot end in dual hole.

1.6 Encoded quantum gates Now consider worldlines of holes.

1.6 Encoded quantum gates Topological quantum gates are encoded in the way worldlines of primal and dual holes are braided.

primal ฀hole smooth฀boundary dual hole dual hole rough฀boundary ฀hole primal primal qubit dual qubit X p Z d 1.6 A CNOT-gate p Z p X d X c X c X t • Propagation relation: X c − → X c ⊗ X t . • Remaining prop rel Z c → Z c , X t → X t , Z t → Z c ⊗ Z t for CNOT derived analogously.

1.6 Topological quantum gates

1.7 Universal gate set • Need one non-Clifford element: fault-tolerant preparation of | A � := X + Y 2 | A � . √ Singular Qubit • FT prep. of | A � provided through realization of magic state distillation ∗ . *: S. Bravyi and A. Kitaev, Phys. Rev. A 71, 022316 (2005).

1.8 Error threshold Perfect error correction: 11% With measurement error: 2.9% 0 . 75% [1] − 1 . 05% [2] Error per gate: • Comparison: Highest known threshold [no geometric constraint] is 3% [3]. [1] R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98, 190504 (2007). [2] D.S. Wang et al., Phys. Rev. A 83 020302(R) (2011). [3] E. Knill, Nature (London) 434, 39 (2005).

1.8 Error Model Error sources : 1. | + � -preparation: Perfect preparation followed by 1-qubit partially depolarizing noise with probability p P . 2. Λ( Z )-gates (space-like edges of L ): Perfect gates followed by 2-qubit partially depolarizing noise with probability p 2 . 3. Hadamard-gates (time-like edges of L ): Perfect gates followed by 1-qubit partially depolarizing noise with probability p 1 . 4. Measurement: Perfect measurement preceeded by 1-qubit partially depolarizing noise with probability p M . • No qubit is ever idle. (Additional memory error - same threshold) • For threshold set p 1 = p 2 = p P = p M =: p .

Overhead Operational overhead at 1 / 3 of error threshold.

Overhead Overhead in Knill’s scheme.

Part II: Twists, Subsystems, and Color

Twists First consider twists in the surface code: H H H H H H Z X o 45 X Z Z Z Z Z X Z X X X X Z X surface code H. Bombin, Phys. Rev. Lett. 105, 030403 (2010).

Twists Z X Z X X Z X Z Z Z X X X Z X Z New lattice: old sites = white faces old faces= black faces

Twists light cycle: measures plaquette stabilizers Z X (included ``magnetic charge’’) X Z Z X dark cycle: X Z measures site stabilizers (included ``electric charge’’)

Twists light cycle: measures plaquette stabilizers (included ``magnetic charge’’) dark cycle: measures site stabilizers (included ``electric charge’’)

Twists was dark face

Twists was dark face

Twists was dark face coloring inconsistent! twist

Twists Strings around twist do not close! String changes color light <-> dark when passing the cut twist

Twists double loop closes

Subsystem codes Example: Bacon-Shor code Stabilizer generators: Z Z Z Z Z Z X X X X X Z Z X X X X X Z Z + translates + translates Encoded Pauli operators: Z Z Z X X X X X Z Z Encoded Z Encoded X

Subsystem codes Example: Bacon-Shor code Stabilizer generators: Encoded Pauli operators: Z Z Z Z Z Z Z Z X X X X X Z X X X X X Z Z X X X X X Z Z Z Z + translates + translates Encoded Z Encoded X Measured operators: * Do not mutually commute X * Commute with stabilizer and Z Z X encoded Pauli operators + translates + translates

Subsystem codes stabilizer n-qubit Pauli group Z Z Z Z Z Z X X X X X Z Z X X X X X Z Z centralizer ( Z ) gauge group Z Z X X X X X Z Z X contains encoded Paulis Z Z X Generated by the meaeasured operators

Subsystem codes centralizer Z gauge group Z ( ) Z X X X X X Z Z X contains encoded Paulis Z Z X Generated by the meaeasured operators acts here acts here gauge qubits system qubits

Color codes Z Z Z Z Z X Z X X X X X • Tri-valent graph • One qubit per site

Color codes Z Z Z Z Z X Z X X X X X • Two stabilizer generators per face, one X + one Z -type • Topology: #encoded qubits = 4 × #handles.

Color • Closed strings represent encoded Pauli operators. • 3 strings can end in a comon vertex. • Must all have different color. Image taken from: H. Bombin, MA. Delgado, PRL 97 (2006).

Part III: Topological color subsystem codes H. Bombin, New J. Phys. 13, 043005 (2011).

TCSCs lattice Λ

TCSCs Q1: What is the gauge group G ? decorated lattice Λ Z 1 Z 2 , Y 2 X 3 ∈ G .

TCSCs Q2: What is the centralizer Z ( G ) ? • Elements of Z ( G ) are consistent shadings of Λ.

TCSCs Q3: What is the stabilizer S ? Z -type stabilizer X/Y -type stabilizer • Two stabilizer generators per (normal) face.

TCSCs Q4: What is a twist? ? Z -type stabilizer X/Y -type stabilizer Yes No • A twist is a face with odd number of edges. • Only one stabilizer generator per twist face.

TCSCs twist • A twist changes the color of an encirling string operator.

TCSCs X Z Encoding of a qubit in 4 twists -1 Entangling gate: qubit 2 qubit 1

TCSCs Entangling gate: X Z -1 qubit 1 qubit 2 What happens to Z 1 : = = Z 1 − → Z 1 ⊗ X 2 • Can implement entire Clifford group by braiding twists.

More exotic codes out there • Tuarev-Viro codes: Universality without state distillation. R. K¨ onig, G. Kuperberg and B.W. Reichardt, arXiv:1002.2816 (2010).

Summary Topological quantum codes are highly suitable for fault- tolerant quantum computation in 2D qubit lattices with nearest-neighbor interaction. Numbers: • Fault-tolerance threshold of up to 1% so far. • Moderate overhead scaling. Suitable systems for realization: • Cold atoms in optical lattices, segmented ion traps, Josephson junction arrays, ...