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 local Nist racetrack ion trap 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 nonzero error threshold ɛ then arbitrarily long quantum computations 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 & urek (1998), Aliferis & Gottesman & Preskill (2005)
Our setting quantum dots superconducting qubits optical lattices segmented ion traps 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 boundary conditions. qubit lattice, spacing a holes in the code plane microscopic scale 2 a macroscopic scale a log(# gates) microscopic view macroscopic view
1.1 The surface code plaquette stabilizer B p One qubit located on every edge site stabilizer As = syndrome at endpoint harmless error harmful error = 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 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 site stabilizer not enforced primal hole plaquette stabilizer not enforced dual hole There are two types of holes: primal and dual. A pair of same-type holes constitutes a qubit.
1.5 Surface code on plane with holes Surface code with boundary: primal qubit dual qubit p d primal hole dual hole primal hole p dual hole d rough boundary smooth boundary -chain cannot end in primal hole, can end in dual hole. -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.
1.6 A CNOT-gate primal qubit primal hole p dual qubit dual hole d p d primal hole dual hole rough boundary smooth boundary c c t Propagation relation: c c t. Remaining prop rel c c, t t, t c 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 := +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% Error per gate: 0.75% [1] 1.05% [2] 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()-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 45 o surface code H. Bombin, Phys. Rev. Lett. 105, 030403 (2010).
Twists New lattice: old sites = white faces old faces= black faces
Twists light cycle: measures plaquette stabilizers (included ``magnetic charge ) dark cycle: 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: + translates + translates Encoded Pauli operators: Encoded Encoded
Subsystem codes Example: Bacon-Shor code Stabilizer generators: Encoded Pauli operators: + translates + translates Measured operators: Encoded Encoded * Do not mutually commute * Commute with stabilizer and encoded Pauli operators + translates + translates
Subsystem codes Generated by the meaeasured operators n-qubit Pauli group gauge group ( ) stabilizer centralizer contains encoded Paulis
Subsystem codes centralizer gauge group ( ) contains encoded Paulis Generated by the meaeasured operators acts here acts here gauge qubits system qubits
Color codes Tri-valent graph One qubit per site
Color codes Two stabilizer generators per face, one + one -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 1 2, Y 2 3 G.
TCSCs Q2: What is the centralizer (G)? Elements of (G) are consistent shadings of Λ.
TCSCs Q3: What is the stabilizer S? -type stabilizer /Y-type stabilizer Two stabilizer generators per (normal) face.
TCSCs Q4: What is a twist? -type stabilizer Yes? /Y-type stabilizer 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 Encoding of a qubit in 4 twists -1 Entangling gate: qubit 1 qubit 2
= TCSCs Entangling gate: -1 qubit 1 qubit 2 What happens to 1 : = 1 1 2 Can implement entire Clifford group by braiding twists.
More exotic codes out there Tuarev-Viro codes: Universality without state distillation. R. König, G. Kuperberg and B.W. Reichardt, ariv:1002.2816 (2010).
Summary Topological quantum codes are highly suitable for faulttolerant 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,...