FAULT-TOLERANT LOGICAL GATES IN QUANTUM ERROR-CORRECTING CODES - PowerPoint PPT Presentation

FAULT-TOLERANT LOGICAL GATES IN QUANTUM ERROR-CORRECTING CODES Fernando Pastawski with Beni Yoshida arXiv:1408.1720 (soon PRA) QEC 2014, Zurich Monday, December 22, 14

ROBUST GATES FROM NOISY ONES Repetition code Transverse gates G x1 g(x1,y1,z1) y1 G z1 x g(x,y,z) y G x2 z g(x2,y2,z2) y2 G z2 • Transverse gates. x3 g(x3,y3,z3) • Benign error propagation. y3 G z3 • Single errors are recoverable. J. von Neumann. In C. Shannon and J. McCarthy (editors) Automata Studies, pages 43--98, Princeton University Press. (1956). Monday, December 22, 14

ERROR PROPAGATION IN TRANSVERSE CIRCUITS Errors only propagate within blocks. Example: Cnot in CSS stabilizer codes. Monday, December 22, 14

THE EASTIN & KNILL THEOREM (2008) • Transversal logical gates are not universal for QC week ending P H Y S I C A L R E V I E W L E T T E R S PRL 102, 110502 (2009) 20 MARCH 2009 Restrictions on Transversal Encoded Quantum Gate Sets Bryan Eastin* and Emanuel Knill National Institute of Standards and Technology, Boulder, Colorado 80305, USA (Received 28 November 2008; published 18 March 2009) Transversal gates play an important role in the theory of fault-tolerant quantum computation due to their simplicity and robustness to noise. By definition, transversal operators do not couple physical subsystems within the same code block. Consequently, such operators do not spread errors within code blocks and are, therefore, fault tolerant. Nonetheless, other methods of ensuring fault tolerance are required, as it is invariably the case that some encoded gates cannot be implemented transversally. This observation has led to a long-standing conjecture that transversal encoded gate sets cannot be universal. Here we show that the ability of a quantum code to detect an arbitrary error on any single physical subsystem is incompatible with the existence of a universal, transversal encoded gate set for the code. DOI: 10.1103/PhysRevLett.102.110502 PACS numbers: 03.67.Lx, 03.67.Pp Don’t panic ! Fault-tolerant computation is still possible. B. Eastin, E. Knill, Restrictions on Transversal Encoded Quantum Gate Sets, arXiv: 0811.4262 [quant-ph] (2008). Monday, December 22, 14

FAMILIES OF QECC (A LOT ABOUT A LITTLE, A LITTLE ABOUT A LOT) COLOR C. & TORIC C. LEVIN-WEN REED-MULLER C. MODELS & QUANTUM LDPC C. DOUBLE Stabilizer codes TQFT codes TQFT codes Topological Topological Subsystem codes GAUGE quantum field theories quantum field theories COLOR (vacua) (vacua) CODES Monday, December 22, 14

LOGICAL GATES FROM LOCAL INTERACTIONS IN TOPOLOGICAL CODES SUPERCONDUCTING OPTICAL LATICES QUBIT ARRAYS SOLID STATE Monday, December 22, 14

ERROR PROPAGATION IN LOCAL CIRCUITS t Errors only propagate geometrically by some constant radius. Monday, December 22, 14

THE BRAVYI-KÖNIG THEOREM (2012) Stabilizer • Under a more physically realistic setting Subsystem TQFT TQFT Logical gate U : low-depth unitary gate (i.e. Local unitary) D-dim lattice Theorem • For a stabilizer code in D dim, logical gates implementable by local circuits are restricted to the D-th level of the Clifford hierarchy. Bravyi, S., & König, R. (2013). Classification of Topologically Protected Gates for Local Stabilizer Codes. Physical Review Letters, 110(17), 170503. Monday, December 22, 14

THE BRAVYI-KÖNIG THEOREM (2012) Stabilizer • Under a more physically realistic setting Subsystem TQFT TQFT Logical gate U : low-depth unitary gate (i.e. Local unitary) D-dim lattice Theorem • For a stabilizer code in D dim, logical gates implementable by local circuits are restricted to the D-th level of the Clifford hierarchy. ??? Bravyi, S., & König, R. (2013). Classification of Topologically Protected Gates for Local Stabilizer Codes. Physical Review Letters, 110(17), 170503. Monday, December 22, 14

CLIFFORD HIERARCHY Gottesman, D., & Chuang, I. L. (1999). Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations, 402(6760), 390–393. P 0 ≡ Sets of unitary transformations on N qubits = Pauli group: X,Y,Z, XX, -ZZIZZI, ... P 1 = Clifford group: CNOT, Hadamard, R 2 ,... P 1 Clifford group is classically simulable. ✓ 1 ◆ 0 P = R 2 = 0 i P 3 has R 3 & Toffoli. Not a group, but a set. P 3 ✓ 1 ◆ 0 R 3 = e i π / 4 0 P n +1 = { U : ∀ V ∈ Pauli , UV U † V † ∈ P n } D. Gottesman (1998), The Heisenberg Representation of Quantum Computers, arXiv:quant-ph/9807006. Monday, December 22, 14

EXAMPLE: COLOR CODE Y = -Y ⨂ N Z = Z ⨂ N • Transverse Gates: X = X ⨂ N H = H ⨂ N Cnot = Cnot ⨂ N P = ⨂ j ∈[ 1,N ] P j±1 Full transverse Clifford group! (assuming logical qubits can be stacked) Bombin, H., & Martin-Delgado, M. (2007). Topological Computation without Braiding. Physical Review Letters, 98(16), 160502. Daniel Nigg, Markus Müller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel Angel Martin-Delgado, and Rainer Blatt. Quantum Computations on a Topologically Encoded Qubit. Science 2014 Monday, December 22, 14

LITERATURE & RESULTS Transverse gate Const. depth circ. + Quantum Code type no geometry locally defined code B. Zeng, A. Cross & I. L. Chuang S. Bravyi & R. König Stabilizer 2007 2013 B. Easting & E. knill Arbitrary 2008 Now! F. Pastawski & B. Yoshida Subsystem 2014 Friday! M. Beverland, R. T. König, F. TQFT Pastawski, J. Preskill & S. Sijher 2014 Monday, December 22, 14

OUTLINE • Cleaning in Quantum error correcting codes • Stabilizer codes • Sub-system codes • Central proof ideas. • Summary of gate constraints. • Conclusions & further directions Monday, December 22, 14

STABILIZER CODES SUBSYSTEM CODES & CLEANING LEMMAS Monday, December 22, 14

PRE-CLEANING LEMMA Errors on a region R (subset of qubits) are detectable iff Correctable regions: ∃ Λ R : ρ = P 0 ρ Λ R Tr R [ ρ ] = ρ ⇒ Tr[ O ρ ] = Tr[ Λ † can be cleaned. R ( O ) ρ ] R = Λ † R ( O ) O ¯ O E. Knill, R. Laflamme, Phys. Rev. A 55, 900 (1997). Monday, December 22, 14

THE STABILIZER FORMALISM • The Pauli group |P| = 4 ( N +1) P = h i, X j , Z j i • A stabilizer subgroup [ g i , g j ] := g i g j g † i g † j = • The code space: Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 Gottesman, D. (1997, May). Stabilizer Codes and Quantum Error Correction. quant-ph/9705052. Thesis @ Caltech. Monday, December 22, 14

CLEANING LEMMA R R = Λ † R ( O ) O O ¯ • For stabilizer codes: O ∈ P ⇒ O ¯ R ∈ P • Bounded support growth for locally defined stabilizer codes. • Union lemma: If two correctable regions don’t share stabilizers their union is correctable. Bravyi, S., & Terhal, B. (2009). A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes. NJP , 11(4), 043029. Monday, December 22, 14

EXAMPLE: 5 QUBIT CODE • Stabilizer group S = h ZIZXX, XZIZX, XXZIZ, ZXXZI i • Detects up to two errors anywhere • Encodes 1 logical qubit ¯ ¯ X = XXXXX Z = ZZZZZ • Suppose we loose second and fourth qubits ¯ ¯ X ≡ ZIXIZ Z ≡ Y IZIY Z 1 Z 2 Z 3 Z 4 Z 5 X 1 X 2 X 3 X 4 X 5 Laflamme, R., Miquel, C., Paz, J. P ., & Zurek, W. H. (1996). Perfect Quantum Error Correcting Code. Physical Review Letters, 77(1), 198. Monday, December 22, 14

SUBSYSTEM CODES Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 Poulin, D. (2005). Stabilizer Formalism for Operator Quantum Error Correction. Physical Review Letters, 95(23), 230504–4. Monday, December 22, 14

SUBSYSTEM CODES Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 4 X 5 X 6 X 7 X 8 X 9 Poulin, D. (2005). Stabilizer Formalism for Operator Quantum Error Correction. Physical Review Letters, 95(23), 230504–4. Monday, December 22, 14

SUBSYSTEM CODES Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 4 X 5 X 6 X 7 X 8 X 9 • A gauge subgroup Hamiltonian (Not necessarily commuting) H ∈ K ( G ) Poulin, D. (2005). Stabilizer Formalism for Operator Quantum Error Correction. Physical Review Letters, 95(23), 230504–4. Monday, December 22, 14

SUBSYSTEM CODES Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 4 X 5 X 6 X 7 X 8 X 9 • A gauge subgroup Hamiltonian (Not necessarily commuting) H ∈ K ( G ) • Stabilizer subgroup, center of G (sign freedom): Poulin, D. (2005). Stabilizer Formalism for Operator Quantum Error Correction. Physical Review Letters, 95(23), 230504–4. Monday, December 22, 14

SUBSYSTEM CODES Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 4 X 5 X 6 X 7 X 8 X 9 • A gauge subgroup Hamiltonian (Not necessarily commuting) H ∈ K ( G ) • Stabilizer subgroup, center of G (sign freedom): • Bare logical operators: Poulin, D. (2005). Stabilizer Formalism for Operator Quantum Error Correction. Physical Review Letters, 95(23), 230504–4. Monday, December 22, 14

SUBSYSTEM CODES Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 4 X 5 X 6 X 7 X 8 X 9 • A gauge subgroup Hamiltonian (Not necessarily commuting) H ∈ K ( G ) • Stabilizer subgroup, center of G (sign freedom): • Bare logical operators: • Dressed logical operators: Poulin, D. (2005). Stabilizer Formalism for Operator Quantum Error Correction. Physical Review Letters, 95(23), 230504–4. Monday, December 22, 14