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Cambridge University Press 978-0-521-89787-7 - Quantum Error Correction Edited by Daniel A. Lidar and Todd A. Brun Index More information

Index

((5, 6, 2)) nonadditive code, 261 accessible information, 580 ((9, 12, 3)) nonadditive code, 261, 275 accuracy threshold, 449, 480 ((10, 24, 3)) optimal code, 261, 268 additive ((n, K, d)) code, 261 classical error-correcting codes, 261 ((n, K, d))q code, 312 quantum error-correcting codes, 241, 263, 270 ((n, K, d))q stabilizer code, 314 Aharonov–Bohm effect, 397 [n, k, d] classical code, 63, 64, 196 alanine, 511, 512, 523 [n, k, d]q classical code, 177, 315, 318, 324, 325, 389, amplitude, 6 393 amplitude damping, 135 [n, k, t] classical code, 63 ancilla, 27, 165, 182, 183, 188, 192, 210, 226, 516 [n, k] random classical code, 280, 281 ancillary qubits, 159, 182, 184, 412, 450, 516, 591, [n, k]q classical MDS code, 326 614 [n, k]q classical cyclic code, 317 angular momentum, 95, 549 [[n, k, d]] code, 59, 169, 263, 279, 334 angular momentum algebra, 84 [[n, k, d]]q code, 312, 325, 326 annihilation operator, 83, 85 [[n, k, r, d]] subsystem code, 170 anticommutation, 186 anticommutator, 28 [[n, k, r, d]]q subsystem code, 178 [[n, 1,r,d]] operator stabilizer code, 205 anyon, 397, 481, 623 [[n, k]] random stabilizer code, 198, 281 approximate quantum error correction, 328 Araki–Lieb inequality, 15 [4, 2, 3] classical quaternary code, 195, 196 atom trap, 6 [4, 2] classical quaternary code, 195 augmented check matrix, 560 [7, 4, 3] classical Hamming code, 65 average Hamiltonian theory, 105, 113, 115, 379 [[4,0,2]] code, 196 [[4,1,2]] code, 328, 332, 333 B92 protocol, 579 [[4,1,3;1]] code, 189, 196 ball, 63 [[4,1;1]] code, 188, 195 billiard, 283 [[5,1,3]] code, 71, 189, 328, 332, 333, 343, tennis, 283 345 Bayes’ rule, 15, 284–286 [[5,1,3]]2 code, 326 BB84 protocol, 563, 563, 564, 566, 579, 580 [[6,0,4]]2 code, 326 BCH (Bose–Chaudhuri–Hocquenghem) code, 318 [[7,0,3]] code, 267 belief propagation, 283, 302 [[7,1,3]] code, 65, 71, 73, 267, 343 Bell basis, 434 [[9,0,3]] graph state, 275 Bell states, 12 [[9,1,3]] code, 61, 70, 275, 343, 345 Bernoulli numbers, 9 [[9,1,4,3]]2 Bacon–Shor code, 178 Berry phase, 397 [[10,0,4]] graph state, 268 billiard ball, 283 [[10,1,3]] code, 268 binary quantum BCH code, 318 [[12,2,3]] code, 298 binary symmetric channel, 280, 286, 570 [[128, 2, 8]] toric code, 464 bit-flip, 47, 54, 195, 541 [[73,1,9]] triangular color code, 477 bit-flip channel, 48 [[85,1,7]] planar toric code, 467 bit-flip code, 59 [[96,4,8]] color code, 475 Bloch space, 8

657

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-89787-7 - Quantum Error Correction Edited by Daniel A. Lidar and Todd A. Brun Index More information

658 Index

Bloch sphere, 6, 7, 208, 529 code mixed states in, 7 additive pure states on, 7 over GF (4), 262 Bloch vector, 7, 8 algebraic, 282 Bohr correspondence principle, 612 as eigenspace of commuting operators, 193, 262 Bohr frequency, 622 Bacon–Shor, 172, 178, 425, 428, 429 Boolean function, 276 BCH (Bose–Chaudhuri–Hocquenghem), 318 bootstrapping, 198 narrow-sense, 319 Born approximation, 616 bit-ﬂip, 173, 217 bound canonical, 191, 194, 573, 574 BCH, 323 classical classical BCH, 318 Gilbert–Varshamov, 280 cyclic code, 317 Hamming, 280 dual of, 63 Singleton, 320 error-correcting, 47 dual, 345 generator matrix of, 63 iterated, 345 Goethals, 271 quantum sparse, 176 Gilbert–Varshamov, 281 Clifford subsystem, 178 Hamming, 60, 77, 178 color, 472, 474, 476, 481, 624 Singleton, 177, 178 concatenation, 172, 173, 279, 504 sphere-packing, 323 convolutional, 182, 200, 231, 232, 234–236, 240, 244, boundary, 455 250, 253, 255, 260, 297 boundary operators, 459 CRSS (Calderbank–Rains–Shor–Steane), 195, 196 bounded control, 351, 374 CSS (Calderbank–Shor–Steane), 43, 65 bounded linear operators, 5, 329 CSS entanglement-assisted, 559, 561, 570 bounded-strength dynamical decoupling, 120 degeneracy, 175, 480 bra, 5 dual, 63, 234, 263, 320, 389 bra-ket notation, 5 entanglement-assisted quantum error-correcting branyon, 481 (EAQECC), 182 Bratteli diagram, 95 error-correcting, 47 breeding protocol, 558, 579 Goethals, 269 CSS, 561, 565, 567, 569 Gray, 389 Hamming, 65, 387 Cauchy sequence, 4 LDPC (low-density parity check), 176, 199, 304 cavity, 85, 547 linear, 61, 199, 269, 280, 315, 389 Cayley graph, 120, 122, 391, 393 local, 456, 478 CDD (concatenated dynamical decoupling), 355, parameters, 299, 325 355–358, 361, 362, 364–366, 374, 375, 535 planar, 472 cell Preparata, 269 dual, 494 primitive, 319 primal, 494 private classical, 581 cellular automata, 159 probabilistic, 282 centralizer, 69, 116, 177, 184 purity, 178 channel quantum amplitude damping, 328, 333, 342 BCH, 318 authenticated classical, 563 error-correcting, 232, 261, 301, 319 binary symmetric, 280, 570 concatenated, 181 depolarizing, 176, 281, 295, 514, 539 degenerate, 72, 89, 163 quantum, 44, 53, 164, 556 distance, 59, 175, 196, 261, 272 channel estimation, 565, 569 error probability, 182 Choi matrix, 331, 339 nondegenerate, 59, 89, 195 classical DFS, 78 Goethals–Preparata, 274 classical information over noisy quantum channel, MDS, 325 581 quaternary, 195, 197 Clebsch–Gordan coefﬁcients, 84 rate, 296 Clifford Reed–Muller, 269, 270, 272, 273 group, 74, 75, 76, 242, 417, 423, 433, 443, 518 Reed–Solomon, 319 transversal, 477 narrow-sense, 319 operation, 244, 421, 423, 424, 481, 505 single-qubit, 225 subsystem code, 178 space, 65–67, 69, 71, 73, 129, 130, 143–145, 170, 176, clique, 268 184, 187, 188, 195, 205, 206, 215, 217–220, closed quantum system, 112 222, 226, 277, 289, 443 CNOT (controlled-NOT) gate, 10 sparse, 279, 297, 298, 300, 304

Index 659

stabilizer, 76, 261 CP (Carr–Purcell) sequence, 105–108, 110, 114, 115, subsystem, 163, 165, 169, 177, 178, 481, 581 118, 123, 356, 377, 530 surface, 159, 461, 465, 468, 479 CP (completely positive) map, 17, 33, 43, 180, 208 three-qubit bit-flip, 207, 554, 576 CPMG (Carr–Purcell–Meiboom–Gill) sequence, 373, 530 topological, 455–458, 461, 472, 475, 480, 481, 518 CPTP (completely positive trace-preserving) map, 3, 34, toric, 464, 467, 480 191, 201, 202, 208, 328, 330 triangular, 477 creation operator, 83 turbo, 200, 292, 293, 295, 297, 305 critical environment, 585 classical, 279, 293 cryptography, 81 quantum, 292, 293, 295, 305, 306 CSS (Calderbank–Shor–Steane), 43 coherence-preserving states, 103 CSW (codeword stabilized codes), 261, 266 coherent information, 280 cumulant expansion, 616 collective curvature, 400 decoherence, 83–86, 95, 96, 102, 103, 119, 406, 407, cycle, 114, 115, 120, 121, 390, 392, 438, 458–460, 465, 514 477, 483, 495, 501 dephasing, 79, 81, 84, 85, 103, 406, 525 cyclic group, 310, 313 dissipation, 103 cyclic property of trace, 332 spin operators, 84 communication complexity, 433 dark state, 401, 403 commutant, 94 Davies generator, 623 commutator, 8, 9, 184, 313, 597 DD (dynamical decoupling), 4, 36, 78, 104, 105, 351, complete positivity, 32 375, 376, 541–543 completely mixed state, 7 bang-bang, 377 completely positive (CP) map, 17, 33, 43, 180, 208 combinatorial, 376 completely positive trace-preserving (CPTP) map, 3, 17, concatenated (CDD), 355–358, 361, 362, 364–366, 34, 191, 201, 202, 208, 328, 330 374, 375, 535 completeness, 16 concatenated Uhrig (CUDD), 365 completeness relation, 24 embedded (EMD), 372 complex conjugate, 5 experimental, 519 complex optimization, 327, 329, 334 naive random (NRD), 372 computational basis, 48, 74, 401 nested Uhrig (NUDD), 367 computational complexity, 348, 612 periodic (PDD), 113, 114, 118, 119, 353, 372, 535 computing with linear optics, 433 quadratic (QDD), 366 random path (RPD), 372 concatenation randomized (RDD), 355, 372 experimental of DFS and QECC, 514 symmetric (SDD), 118, 373 for fault tolerance of holonomic QC, 428 symmetric random path (SRPD), 373 of a convolutional code, 295 Uhrig (UDD), 357, 360, 361, 363, 365, 366, 368, 373, of bit-flip and phase-flip codes, 70 530 of classical codes, 173 decoder of convolutional code, 292 bit-wise, 285 of DD sequences, 366 maximum-likelihood, 287 of DFS and QECC, 102, 104 soft, 286 of strings, 568 decoding, 47, 48, 145, 170, 176, 190, 231, 237, 257, 260, conditional entropy, 15 279, 282, 284, 285, 287, 289, 293, 303, 513 conditional probability, 15, 285, 303 in exponential time, 282 connection, 400 in polynomial time, 282 u (1)-valued, 408 decoherence, 3, 42, 76, 104, 108, 158, 202, 213, 224, 327, u (3)-valued, 403 331, 360, 380, 522, 533, 585, 618 adiabatic, 413 control, 106, 204, 520 form, 408, 409 estimates of, 117 irreducible, 400 decoherence problem, 46 non-Abelian gauge, 407 decoupling scheme continuous-variable system, 554, 571, 581 bang-bang, 379, 380 control cycle, 119, 380, 526 bounded strength, 377 control propagator, 115, 120, 380, 392 Eulerian, 391 control sequence, 520 periodic, 377 Cooper pair, 526 regular, 380 correctable errors, 59, 166, 193, 203, 211, 277, 554 degeneracy, 82, 289, 398 correlation function, 107, 141, 360, 540, 585, 616 density matrix, 5 two-point, 587, 599 dephasing errors, 43 correlations depolarizing channel, 281 Dirac-delta, 616 depolarizing noise, 44 coset, 64, 265, 315, 460 deviation density matrix, 511

660 Index

DFS (decoherence-free subspace), 4, 78, 78–92, 94, entropy, 13 96–99, 101–104, 106, 127, 163–165, 179, 180, binary, 570 346, 406, 412, 514, 523, 525, 543, 602 of entanglement, 14, 38 for atoms in a cavity, 85 Shannon, 280 for collective decoherence, 102 von Neumann, 13, 14, 22 for collective dephasing, 81 environment, 578, 585, 588, 596, 605 for stabilizer errors, 88 EPR pair, 182, 198 NMR experiment, 514 error diamond norm, 132, 614 classical, 62, 544 dibromothiophene, 523, 525 map, 616 difference matrix, 384, 391 model, 33, 41, 45, 220, 294, 480, 497, 539, 602 dimension of a vector space, 61 changes under concatenation, 176 dipolar coupling, 532 propagation in fault-tolerant quantum circuits, 619 Dirac notation, 5 rate, 225, 287, 299, 403, 482, 505, 514 discord, quantum, 12, 15, 16, 18, 26, 339 rate reduction, 225 discretization of errors, 42 syndromes, 237, 276 distance threshold, 177, 428, 470, 480 Kolmogorov, 23 error-avoiding codes, 103, 164 trace-norm, 23 Euler distillation characteristic, 458 entanglement, 182, 553, 554, 579 function, 610 entanglement-assisted entanglement, 558 Eulerian stabilizer entanglement, 556 circuit, 121 divisor, 322 cycle, 121, 124, 392, 393 drift Hamiltonian, 111, 377 graph, 121 dual exchange gate, 545 bound, 345 exchange interaction, 102 distance, 323, 389 expectation value, 19 feasible point, 345 exponential resources, 291 lattice, 464, 466 extended gadget, 146–154 optimization, 336 bad, 150 dual of a classical linear code, 63 good, 150 dynamical exponent, 587 extraspecial p-group, 310 dynamics in the bang-bang limit, 108 Dyson series, 8–10, 23, 586–588, 602 factoring, 553 fault path, 132, 134, 137, 147, 154, 449 EAQECC (entanglement-assisted quantum bad, 152, 449, 450 error-correcting codes), 182–184, 186–191, expansion, 133, 150 194–196, 198–200 good, 152, 449, 450 ebit, 182, 183, 186, 188, 191, 198, 254–256, 553, 558, sum over, 134 562, 567 trivial, 152 logical, 555, 558, 562 fault-tolerant design, 450 physical, 554, 558, 562 fault-tolerant operation, 443 effective Hamiltonian, 9, 82, 116 Toffoli gate, 417 EigQER (eigen quantum error recovery), 341, 343 Fermi Golden Rule, 617 electron spin, 373, 535, 542 Feynman path integral, 397 encoded basis, 204, 217 fiber bundle, 408 encoding operation, 182, 188, 250, 330, 346 Fibonacci scheme, 159 energy eigenstate, 136 fidelity ENIAC, 126 average entanglement, 331, 336 ensemble, 34, 169, 283, 329, 405, 523 entanglement, 329, 331, 333, 343, 513, 514, 524, 526 entanglement, 12, 13, 25, 35, 39, 76 maximum output, 24 distillable, 334 minimum, 329, 336 entropy, 13 quantum, 327 fidelity, 112 Uhlman, 23 generation, 550 filter function, 358, 530 measure, 14 finite field, 177, 271, 307, 309, 388 pseudo, 374, 535 first homology group, 460, 480 relative to choice of observables, 13 Fourier expansion, 363, 364 shared, 182, 198, 561 Fourier transform, 107, 359, 360, 381, 576, 577, 616 entanglement distillation protocol yield, 556 discrete, 381 entanglement-assisted convolutional codes, 253 free-field theory, 587 entanglement-assisted quantum error correction, Frobenius norm, 22 189 fullerene molecule, 528

Index 661

Gacs bare, 614 and noisy cellular automata, 159 decoupling, 390 and reliable classical computation, 159 dynamics, 18 gadget engineering, 106 bad, 148 internal molecular, 522 extended, 146, 152 interpolating, 419 fault-tolerant, 127, 147 physical, 614 for measuring an observable, 145 system, 377 for pure state preparation, 144 Hamming good, 146 bound, 60 measurement extended, 147 code, 62 noisy, 146 distance, 388 preparation, 147 weight, 177, 315 unitary-gate, 147 hard cutoff, 358 gadgets hash function, 580 and organs, 143 hashing bound, 581 and rectangles, 143 Heisenberg coupling, 522 and robustness to noise, 146 Heisenberg uncertainty principle, 242, 563 gauge field, 407, 587 Hermitian map, 18 gauge qubits, 206 Hermitian operator, 5 Gaussian Hilbert space, 4 noise, 159, 540 quiet corner of, 78, 90 random variable, 42, 530 separable, 13 generalized measurement, 24, 33, 35, 129, 144 Hilbert–Schmidt norm, 22 generator matrix, 62, 65, 233, 244, 266, 284, 389 Hilbert–Schmidt inner product, 330, 380 of classical linear code, 63 history of syndromes, 591, 605 genus, 457 holonomy, 399–401, 414–416, 425 GF (4), 195, 262 group, 400 Gibbs inverse, 400 distribution, 287 map, 399 state, 621 homology, 458, 460–462, 480 Gilbert–Varshamov bound, 280 HQC (holonomic quantum computation), 398 global phase, 6, 359, 420, 571 Hubbard model, 547 Goethals code, 269, 271 hypercube, 456, 595, 598, 619 Goethals–Preparata code, 272 hyperfine good coding scheme, 282 field, 528 Gram–Schmidt procedure, 185 interaction, 527, 588 Gram–Schmidt symplectic orthogonalization procedure, level, 130 559 splitting, 130 graph state, 6, 515 Cayley, 120, 122, 391, 393 hyperfine-induced dephasing, 527 Eulerian, 121 Tanner, 199, 297–299, 302–305 independent generators, 185 Grassmann manifold, 410 inequality Green functions, 592 Araki–Lieb, 15 ground state, 39, 287 Lieb–Ruskai, 15 group linear matrix, 332, 346 Abelian, 174, 187, 262, 300 information reconciliation, 565 centralizer, 116 information theory, 76 Clifford, 75, 76, 242, 417, 423, 433, 443, 518 initialization-free finite, 120, 309 decoherence-free subspace, 90 generators, 67, 121 noiseless subsystem, 98 homomorphism, 459 inner product space, 4, 22 index, 381 interference, 46, 108, 127, 136, 547, 607 non-Abelian, 66, 90, 188 interleaver, 292–297 normalizer, 265 ion trap quantum computer, 514 quotient, 69, 310, 460 irreducible representations (irreps), 82 stabilizer, 263, 289, 311 Ising model, 286, 287, 469, 471, 623 subgroup, 74, 262 and decoding, 287 group theory, 120, 313 for toric code, 470 ground state, 287 Hadamard matrix, 382, 433 phase diagram, 470 Hadamard transform, 65 random three-body, 479 Hamiltonian, 8 random bond, 469

662 Index

isometry, 143, 169, 316, 317, 340, 450 Born–Markov, 405 isotropic subgroup, 185, 196 discrete, 29 iterated bound, 345 Heisenberg picture, 622 iterative optimization, 334 Lindblad, 28, 30, 32, 41, 202, 620 itinerant electrons, 588 Markovian bit-ﬂip, 207 Nakajima–Zwanzig, 220 Jamiolkowski isomorphism, 330, 332, 346 post-Markovian, 32 Jaynes–Cummings Hamiltonian, 85, 103, 548, 549 quantum, 26 Johnson–Nyquist noise, 588 stochastic, 41 Josephson junction, 526, 588 time-convolutionless, 220, 615 time-local, 31 Keldysh matrix algebra, 95 contour, 592, 594, 606, 608 maximally entangled, 14 formalism, 592 maximum-likelihood decoding, 285 ket, 5, 292 MDS (maximum distance deparable) code, 325 Kitaev model measurement four-dimensional, 613, 623 continuous, 203, 213 three-dimensional, 623 current, 214 two-dimensional, 622 fault-tolerant, 417 Kondo problem, 588 projective, 20, 33, 132, 340, 451 Kraus operator, 16 weak, 203, 208, 216 block-diagonal representation, 86 memory kernel, 32 sum representation, 16, 17, 28–30, 58, 338 metastable observable, 621 Kronecker product, 12 methyl group, 523 metric, 112, 316 Lamb shift, 615 metric space, 4 Hamiltonian, 30 minimum distance decoder, 283 lateral quantum dot, 588 mirror lattice plane, 533 D-dimensional, 601 spherical, 533, 534 for Bacon–Shor code, 172 mixed state, 5, 6, 19, 136, 329, 539 for Kitaev models, 622 rank, 7 triangular, 479 MOOS (mutually orthogonal operation set), 353 Union Jack, 479 mutual information, 15, 167, 280 law of large numbers, 564 classical, 15 leakage error, 91, 92, 102, 141, 403 quantum, 15 leakage space, 141, 142 level reduction, 154 Nakajima–Zwanzig Lie algebra, 82–84, 95, 400 master equation, 220 Lieb–Ruskai inequality, 15 projection operator technique, 26 lift of a curve, 408 narrow-sense limited thermalization, 102 BCH code, 319 Lindblad generator, 32, 202, 218 Reed–Solomon code, 319 Lindblad–Gorini–Kossakowski–Sudarshan master nice error basis, 381, 391 equation, 620 Nishimori Lindbladian, 202, 203 line, 470, 471 LMI (linear matrix inequality), 332, 346 temperature, 286, 287, 303 local leakage noise, 141 NMR (nuclear magnetic resonance), 105, 356, 430, 510 local Markovian noise, 134 experimental quantum error correction, 514 locality, 117, 130, 132, 139, 142, 227, 377, 386, 387, 430, no-cloning theorem, 47, 51, 563 455, 456, 465, 476, 480, 614, 622 noise, 33, 45, 126, 130, 132, 135, 138, 155, 206, 287, 361, LODD (locally optimized dynamical decoupling), 531 433, 540, 568, 614 log det heuristic, 347 classical, 46, 373, 539 logarithmic time scale, 613 collective, 164 logic gates, 181, 397, 618 depolarizing, 176 fully coherent, 138 magnetic resonance imaging, 535 Gaussian, 159, 540 magnetization, 287 Johnson–Nyquist, 588 Magnus expansion, 9, 9, 10, 113, 114, 119, 549, 603 non-Markovian, 138 malonic acid, 373 power spectrum, 110, 526, 530 map, 17 spectrum, 358, 361, 544 Markovian dynamics, 32, 619 noiseless subsystem, 88, 93–95, 97, 98, 101, 179 deﬁning feature, 27 noisy quantum channel, 198 master equation, 26 non-CP maps, 222

Index 663

non-Markovian dynamics, 27, 204 phase norm, 112, 133, 370, 614 coherence, 520 diamond, 25, 132, 614 decoherence, 511 Euclidean, 22 flip, 54, 174, 515 Frobenius, 22 gate, 187, 243, 422 Hilbert–Schmidt, 22 randomization, 530 operator, 22 phonon trace, 22, 43, 137 acoustic, 588 unitarily invariant, 22 bath, 83 normal ordering, 597, 609 induced decoherence, 405 normalizer, 69, 74, 75, 264, 265, 272 wavelength, 84 NP-complete problem, 281, 286 photodetector, 39, 578 NP-hard problem, 268, 283, 286, 347 photon polarization, 164 NS (noiseless subsystem), 78, 78, 95, 96, 99, 101, 102, Planck constant, 8 104, 163–165, 179, 180, 406, 407 Plotkin sum, 271 initialization-free, 98 Poisson process, 32 nuclear spin, 130, 355, 522, 529, 539, 543, 588 polarization, 401, 518, 533 NUDD (nested Uhrig dynamical decoupling), polynomial time 367 decoding, 282, 570 solution, 468, 612 observable, 19 population relaxation, 358, 366 expectation value of, 19, 215 positive operator, 17, 346 metastable, 623 positive semidefinite operator, 5, 331 OPE (operator product expansion), 595, 597, postselection, 17, 159, 518 605–608 POVM (positive operator valued measure), 20, 21, 23, 24, operation element, 16 27, 38 operator power spectrum, 107, 110, 526, 530 boundary, 463, 478 Preparata code, 269 density, 40, 205 preparation gadget, 147 ebit stabilizer, 558 prime number, 177 error, 43, 57, 207, 377, 386, 555 primitive code, 319 logical, 70, 75, 170, 172, 174, 258, 264, 288, 464, 475, principal bundle, 408, 410 476, 484, 490, 558 privacy amplification, 565, 569 Pauli, 10, 59, 68, 69, 124, 186, 195, 244, 262, 303, 383, probability leakage, 17 449, 472 projection superoperator, 116 projection, 16, 98, 340 projective representation, 116, 381 string, 464–466, 474, 475 projective syndrome measurement, 340 time-ordering, 9, 352, 359 projector, 16, 19, 133, 290, 414, 419, 557 unitary, 136 pulse shaping, 368 operator algebras, 93, 163 pure dephasing, 117, 355, 358, 366, 602 operator basis, 352 pure state, 5, 6, 87, 336, 510 operator norm, 22 purification, 175, 280, 450 order of decoupling, 357 entanglement, 183, 551 orthogonal purity, 20, 216, 323, 346, 347, 521, 534 arrays, 377, 386, 388, 390 complement, 86, 205 QDD (quadratic dynamical decoupling), 366, 367 decomposition of the identity, 19 QECC (quantum error-correcting code), 43 orthogonality constraint, 341 quantum check matrix, 189, 190, 240, 256–257, 258, outer product, 6 559 Overhauser field, 588 quantum circuit, 128, 138, 231, 495 quantum codeword, 397 pair-interaction Hamiltonian, 377, 387 quantum communication, 181, 199, 231, 254, 553, 571, parameter drift, 113 578, 579 parity check, 196, 297 quantum computer architecture, 536 parity check matrix, 63, 193, 574 quantum convolutional coding, 232 parity kicks, 110 quantum discord, 12, 15, 16, 18, 26, 339 partial matrix element, 17, 100 quantum dots, 398, 578 partial trace, 13, 330 quantum error correction, 43, 46, 48, 56, 65, 76, 77, 90, path ordering, 399 126, 144, 160, 169, 199, 327, 509, 510, 512, Pauli group, 66, 67, 69, 74, 75, 169, 262, 288, 417, 571, 514, 516, 517, 537, 553, 554, 578, 579, 585, 604 596, 609, 613 Peierls argument, 623 continuous-time, 202, 206 perfect matching algorithm, 468 continuous-variable, 577 permutation symmetry, 79, 83, 103, 164 operator, 163, 413

664 Index

quantum gate, 11, 76, 131, 335, 403, 545, 588 SARG04 protocol, 579 π/8, 417, 423, 480 scalar CNOT, 11, 242, 417, 480, 500 coupling, 522 controlled-phase, 243 Liouville product, 621 Hadamard, 53, 208, 259, 420, 464 operator acting as, 82, 166 identity, 482 product structure, 408 NOT, 10, 547 state transforming as, 84 SWAP, 11 system–bath interaction, 84 quantum information theory, 17, 183, 563 scaling dimension, 587, 600 quantum jumps, 37 Schmidt decomposition, 14, 344 quantum key Schrodinger¨ equation, 8 distribution, 563, 564, 566, 580 Schur product, 383 entanglement-based, 566 SDP (semideﬁnite program), 332–334, 336, 338, 340, expansion, 553, 554, 562, 564, 565, 569, 570, 580 342, 344, 345, 347 quantum memory, 103, 352, 530, 613, 619 secret key, 553, 562–568 quantum money, 563 distillation, 567, 568 quantum mutual information, 18 generation, 567 quantum operation, 42, 89, 166, 203, 291, 328, 442, 537, uncompromised, 569 564 security quantum phase transition, 599, 600 analysis, 580 quantum process tomography, 4 criterion, 566, 580 quantum register, 119 of quantum key expansion, 562 quantum serial-turbo coding, 232 security proof, 553, 554, 559, 561, 563, 565, 579, 580 quantum state, 5 Lo–Chau, 580 quantum trajectory, 34 Luo–Devetak, 580 qubit, 6, 21, 83, 130, 158, 188, 238, 404, 511, 601 Shor–Preskill, 580 charge, 526, 588 self-correcting quantum memory, 620 dark state, 401 separable electron, 528 Hilbert space, 5, 13, 408 ﬂux, 526 state, 12, 15, 16 harmonic oscillator representation, 140 seven bridges of Konigsberg,¨ 121 in fullerene, 528 Shannon Josephson junction, 526, 588 capacity, 280, 570 memory, 543 entropy, 14 nuclear, 528 shift register, 234, 235, 244 proton, 523 classical, 231, 235 spin, 521 quantum, 245, 247, 249–251, 253 trapped ion, 530 shifted symplectic product, 239, 240, 254 qudit, 379–381, 383, 386, 387, 389, 392, 393, 430 matrix, 240, 241 maximally entangled, 557 Shor code, 61, 70, 275, 343, 345 quiet corner of Hilbert space, 78, 90 continuous-variable, 581 qutrit, 430 sifting, 564 sign matrix, 383 Rabi single photons, 482, 517, 520 coupling, 401, 549 single-qubit operation, 206, 212, 405, 427, 440, 568 frequency, 403, 405 Singleton bound, 320, 325, 326 oscillation, 528, 529 singular values, 14, 24, 338, 341 Ramsey interference, 526 soft high-frequency cutoff, 358, 531 random coding soft pulse, 510 quantum, 280 spectral decomposition, 7, 17, 415 Shannon’s, 279 spectral density, 108, 359, 405, 617 rank, 5 spin-boson model, 83, 103, 357, 358, 588, 601 raw key, 564, 565 spin-glass, 286 RDD (randomized dynamical decoupling), 355, 372 spin-orbit coupling, 405, 546, 548 recovery superoperator, 57 spontaneous emission, 40 reduced density matrix, 13, 14, 131, 220, 221, 359, 405, stabilizer code, 65, 311 591, 592 centralizer, 69 Reed–Muller code, 269, 270, 272, 273 circuit for encoding into, 74 Reed–Solomon code, 319, 320 cyclic code, 271 refocusing, 106, 107, 127, 510, 526, 530 degenerate errors, 284 relative phase, 533, 534 distance, 261 renormalization group, 596, 619 entanglement-assisted, 253 representation theorem for noiseless subsystems, 93 normalizer, 262 ribbon, 295 quantum convolutional, 231

Index 665

union, 263, 265 open, 457 via privacy amplification, 581 orientable closed, 457 stabilizer formalism, 76, 169, 182, 199, 262, 579, 581 topology, 457 stabilizer generator, 68, 71, 72, 169, 172–175, 196, 208, SVD (singular value decomposition), 14, 338, 341 217, 288, 290, 297, 301, 456, 466, 478 SWAP gate, 11, 545 stabilizer group, 67–69, 75, 76, 189, 190, 263, 265, 289, symmetrization 300, 311, 312, 478 in dynamical decoupling, 116, 122, 123, 352, 355, normalizer of, 75 393 standard form of noise channels, 570 for codeword stabilized codes, 266 symmetry and conservation laws, 78 for group generators in EAQECC, 186 symmetry-breaking perturbations, 101 for semidefinite program, 334 symplectic for stabilizer code generators, 194 inner product, 189, 262 Stark shift, 615 partners, 186 state space, 110, 309, 407 product, 252, 560, 572 state vector, 399, 409, 410 representation, 189, 196, 244 statistical model, 470 subgroup, 185 Steane code, 54, 65, 71, 75, 76, 181, 267, 270, 343 syndrome, 63, 72, 144, 148, 171, 175, 176, 205, 211, 261, Stiefel manifold, 410 277, 284, 288, 290, 304, 341, 468, 567, 604 Stinespring representation, 25, 167 extraction, 455, 603 stochastic qubit, 148, 205, 206, 211, 227 differential equation, 203, 214 reduced, 191, 193, 194 equation, 41 syndrome measurement, 73, 78, 129, 144–146, 148, 163, error model, 32 178, 289, 290, 340, 341, 432 errors, 4, 32, 33, 37, 38, 41, 587, 598, 599 system–bath coupling jump, 33 diagonal, 355 master equation, 41 off-diagonal, 355 Schrodinger¨ equation, 39–41 system–bath Hamiltonian, 94, 101, 351 variable, 37 stochasticity, 449 Tanner graph, 199, 297–299, 302–305 string-net, 476, 477 TCL (time-convolutionless), 220 strong Church–Turing thesis, 612 teleportation, 183, 435, 436, 438, 442, 451, 542 submultiplicativity of unitarily invariant norms, 22, 23 for implementing quantum error correction, 160 subsystem principle, 226 gate, 160, 432, 434, 435, 443, 451, 501, 550, 551 superconducting qubits, 130 model for quantum computation, 432–434, 437, 451 superdense coding, 183, 191 one-bit, 435, 436, 451 super-Ohmic, 405 state, 434 superoperator, 5, 53, 55, 56, 58, 61, 121, 131, 133–135, tennis ball, 283 165 tensor product, 12 for amplitude damping, 135 theorem for Eulerian dynamical decoupling, 123, 124 Baker–Campbell–Hausdorff, 572 projection, 116, 201 fixed point, 180 recovery, 57 Margolus–Levitin, 618 unital, 115 Neumark’s, 38 unitary, 615 no-cloning, 47, 563, 579 superoperators quantum capacity, 581 commuting, 615 Stinespring, 167 composition of, 131, 132 Stokes’, 409 distance between, 24 Stone–von Neumann, 575 extensions of, 25 threshold, 155, 157, 431, 449–451, 598–600, 609, 613, limitations of, 136 614, 617 norm for, 25 Wick’s, 141, 587, 593, 598, 599, 605 superposition, 39, 64, 65, 127, 341, 450, 462, 515, 518, thermal coherence length, 591 613 thermal equilibrium, 107, 471 superpropagator, 615 threshold condition, 451 surface time-homogeneous, 27, 28 closed, 457, 478 time-local code, 159, 461, 464, 465, 468, 475, 478 generator, 26 connected orientable closed, 457 master equation, 26, 27 correlation, 484, 486 time-ordering operator, 9 first homology group, 458, 460 topological code, 455–458, 461, 472, 475, 480, 481, 518 genus, 457 topology, 304, 399, 457, 460 of a defect, 496 toric topology, 622 of operators, 485 torus, 455–458, 460, 464, 472, 474, 475

666 Index

trace universal quantum computation alternating form, 316 on a DFS, 104, 406, 525 form, 316 experimental, 525 trace inner product, 380 on a stabilizer code, 76 trace norm, 22 using gadgets, 159 trace-norm distance, 23 via linear optics, 517 between pure states, 137 universality transversal computational, 399 gate, 181, 481, 538 encoded, 400, 407 P , 477, 479 in holonomic quantum computation, 400 π/8, 481 of 2-local Hamiltonians, 430 adiabatic, 423 of DD for time-dependent Hamiltonians, 369 Clifford, 421, 472 of NUDD, 368 CNOT, 416, 417, 425, 472, 481 of QDD, 367 Hadamard, 464, 477 of two-qubit measurements, 438 Toffoli, 428 of UDD, 358 operation, 227, 412, 416, 417, 424, 471 transversality, 479, 481 van Hove limit, 620 turbo vanishing discord, 15 code, 279, 292, 293, 295, 297, 305, 306 virtual degrees of freedom, 94 decoder, 294, 295 von Neumann Turing machine, 138, 599 and classical fault tolerance, 74, 142 two-universal hash function, 565 and noisy cellular automata, 159 von Neumann entropy, 13 UDD (Uhrig dynamical decoupling), 357, 362 uniﬁcation of methods for overcoming quantum errors, 78 wavefunction, 5 union stabilizer code, 263, 264, 265, 268–270, 274–277 weak coupling limit, 620 unit vector, 8 weak coupling regime, 615 unital channel, 164, 180 weak measurement, 20, 22, 31 unitarily invariant norm, 22, 113, 140 weight enumerator, 282 unitary Werner state, 15 error basis, 380 Wiener increment, 214 operator, 8 freedom, 164 Young universal diagram, 97 dynamical control protocol, 369 tableaux, 97 dynamical decoupling, 118, 354 electric computer, 126 Zeeman fault-tolerant quantum computation, 423 interaction, 521, 539 holonomic control, 403 magnetic ﬁeld, 527 initial state, 441 splitting, 105 recovery operation, 166 terms, 377, 532 set of quantum gates, 128, 416, 437, 443, 444, 491, Zeno 542, 614 effect, 4, 36, 110, 219, 220, 357 set of quantum operations, 481 regime, 204, 219, 220, 225, 226 set of weak operations, 227 time, 225