Experiments with and Applications of the D-Wave Machine

Transcription

1 Experiments with and Applications of the D-Wave Machine Paul Warburton, University College London

2 1. Brief introduction to the D-Wave machine 2. Black box experiments to test quantumness 3. Black box experiments to measure the lengthscale of entanglement 4. Application: decoding classical error-correction codes 5. Conclusions

3 Ising Hamiltonian with Transverse Field problem Hamiltonian i i, j Inputs: h i, J ij Outputs: σ z i NP-complete 10 A(t) B(t) annealing schedule H(t) = A(t) Σi transverse field σ i x + B(t) H I problem Hamiltonian energy (GHz) k B T normalised time, t/t f

4 Quasi-spins are SQUID flux states Qubit j Qubit i programmable inductive coupler Multiplexed flux biases provided by on-chip (classical) flux logic DACs

5 D-Wave Vesuvius chip at USC 503 bits 1415 couplers 3,900 DACs 96,000 Josephson junctions 56 wires down cryostat T = 18 mk

6 1. Brief introduction to the D-Wave machine 2. Black box experiments to test quantumness 3. Black box experiments to measure the lengthscale of entanglement 4. Application: decoding classical error-correction codes 5. Conclusions

7 Is there really any quantum mechanics? Vinci, Albash et al. arxiv 1403:4228

8 Qubit Spectroscopy Lanting et al. PRX (2014) 8 qubits Spectroscopy by coupling to ancilla qubit

9 Test of Quantumness: UCL collaboration with USC Boixo et al. Nature Comms (2013) J = 1 (ferromag.) h = 1 h = -1

10 Test of Quantumness: UCL collaboration with USC Boixo et al. Nature Comms (2013) 17-fold degenerate ground state: J = 1 (ferromag.) h = 1 outer spins core spins Single-spin-flip transitions between a cluster GS and the isolated GS are only possible via an excited state h = cluster states, linked by single spin-flip single isolated state, four or more spin-flips distant

11 Test of Quantumness: UCL collaboration with USC Boixo et al. Nature Comms (2013) 17-fold degenerate ground state: J = 1 (ferromag.) h = 1 Hamming distance from Isolated state 8 outer spins core spins 7 h = cluster states, linked by single spin-flip Single-spin-flip transitions between a cluster GS and the isolated GS are only possible via an excited state 4 0 single isolated state, four or more spin-flips distant

12 Rescale the Energy of the Ising Hamiltonian H(t) = A(t) Σi transverse field σ i x + α B(t) H I problem Hamiltonian energy (GHz) k B T A(t) α B(t) α = 1 α = 0.31 α = 0.12 increasing role of thermal fluctuations normalised time, t/t f

13 Experimental Results prob. (isolated state) prob. (single cluster state) N = A(t x ) = 0.31B(t x ) = k B T QUANTUM CLASSICAL Ising energy scale α Increasing effective temperature Vinci, Albash et al. arxiv 1403:4228

14 Comparison with Theory Data and quantum model Classical models 2 prob. (isolated state) prob. (single cluster state) Ising energy scale α O(2) spin dynamics / SA hybrid ( SSSV ) O(3) spin dynamics Ising energy scale α SSSV: Shin, Smith, Smolin, Vazirani: arxiv

15 Experimental Results N = 8 N = 12 N = 16 N = 20 prob. (isolated state) prob. (single cluster state) N = 20 N = 16 N = 12 N = Ising energy scale α QUANTUM CLASSICAL

16 Experimental Results α crossover Quantum-classical crossover at lower effective temperature for larger systems number of qubits N Loss of adiabaticity and/or limited lengthscale of coherence

17 . A classical model that fits: Data and quantum model O(2) spin dynamics / SA hybrid ( SSSV ) Uncertainty in values of programmed h s 2 prob. (isolated state) prob. (single cluster state) Ising energy scale α Ising energy scale α Vinci, Albash et al. arxiv 1403:4228 Shin et al. arxiv

18 Population of the Cluster Ground States: D-Wave Vesuvius Symmetry 16 cluster states should be equally populated N = 8 N = 8 Counts/multiplicity (4;1) (5;4) Isolated state (6;4) (6;2) (7;4) h i = 1 J ij = 1 (8;1) Counts/multiplicity (4;1) (5;4) Isolated state (6;4) (6;2) (7;4) (8;1) h i = 1 J ij = (Hamming distance from isolated state; multiplicity) (Hamming distance from isolated state; multiplicity)

19 Population of the Cluster Ground States: Master Equation Probability/multiplicity D-Wave No calibration correction isolated state (8;1) (7;4) (6;2) (6;4) (4;1) (5;4) Master equation h = J Ising energy scale α Ising energy scale α Ising energy scale α Master equation With crosstalk

20 Population of the Cluster Ground States: SSSV DW no offset SSSV no offset SSSV, h = 0.97 J SSSV, h = 0.94 J SSSV with crosstalk

21 Interim Summary Open system quantum master equation describes well: suppression of isolated state at high Ising energy scale enhancement of isolated state at low Ising energy scale distribution of all ground states for all values of Ising energy scale At present no fully classical model can do all this Suppression of isolated state persists to 40-spin system.. are all the spins entangled??

22 1. Brief introduction to the D-Wave machine 2. Black box experiments to test quantumness 3. Black box experiments to measure the lengthscale of entanglement 4. Application: decoding classical error-correction codes 5. Conclusions

23 Lengthscale of Entanglement Lanting et al. PRX (2014): Entangled single unit cell (8 spins) Are these two spins really entangled??

24 Matrix Product States to Quantify Lengthscale of Entanglement χ = Schmidt rank Crowley et al. arxiv: sub-space accessible with finite χ full Hilbert space (large χ)

25 Experimental Strategy One-dimensional spin chain of length 100, Chimera-compatible 500 random instances: J ϵ ± {0.2, 0.4, 0.6, 0.8, 1} h ϵ ± {0, 0.2, 0.4, 0.6, 0.8, 1} Find ground state by exhaustive search Simulate annealing using Langevin equation (T = 0) over matrix product states with Schmidt rank χ = 1, 2, Categorise each instance by minimum Schmidt rank required to find correct ground state (deterministic) Run same instances on D- Wave 1000 times Categorise each instance by probability of finding correct ground state

26 Experimental Results: 1-D chain of 100 spins Instances not solved with χ = 1 solved with χ = 1 D-Wave Success Probability All instances solved by MPS representation with Schmidt rank of 1 or 2 Some (weak) correlation with D-Wave. thermal effects on D-Wave. need 2-D MPS solver Crowley et al. arxiv:

27 1. Brief introduction to the D-Wave machine 2. Black box experiments to test quantumness 3. Black box experiments to measure the lengthscale of entanglement 4. Application: decoding classical error-correction codes 5. Conclusions

28 Classical Error Correction TRANSMIT noise Source data N bits M bits M bits RECEIVE encoding Communication channel decoding N bits M N redundant bits (e.g. parity check bits) Received data Rate of code, R N / M < 1

29 Ising Code = Parity Checking Sourlas, Nature (1989) e.g. two-bit word mapped onto two spins: Bit A Bit B Parity Bit Parity bit = A + B mod 2 σ A = 2A 1 σ B = 2B 1 Spin A Spin B J AB J AB = σ A σ B

30 Ising Code: Transmit the Couplers Sourlas, Nature (1989) TRANSMIT noise σ 1, σ 2, σ N Source data N spins M couplers M couplers RECEIVE encoding Communication channel decoding J 12, J 13,. J (N-1)N ~ ~ ~ N spins ~ σ 1, ~ σ 2, ~ σ N J 12, J 13,. J (N-1)N Received data ~ Maximum likelihood decoding: find the ground state of spin glass J

31 Ground State Decoding Spin Space Coupling Space

32 Ground State Decoding Spin Space Coupling Space channel noise

33 Ground State Decoding Spin Space Coupling Space channel noise find groundstate

34 Ground State Decoding Spin Space Coupling Space channel noise

35 Ground State Decoding Spin Space Coupling Space channel noise

36 Finite-Temperature Decoding Rujan PRL (1993) Spin Space Coupling Space sample at finite temp. channel noise Maximum entropy decoding: decode at the Nishimori temperature (Requires prior knowledge of channel noise characteristics)

37 Decoding and Quantum Fluctuations Otsubo et al. PRE (2012) Overlap = 1 2 x bit-error-rate T Nishimori At temperatures below the Nishimori temperature, quantum fluctuations can help Unknown: scale of thermal and quantum fluctuations on D-Wave chip

38 Ising Decoding on the D-Wave Vesuvius Chip TRANSMIT noise σ 1, σ 2,. Source data 503 spins encoding: J AB = σ A σ B J ij = couplers J 12, J 13,. binary symmetric channel 1415 couplers J ~ 12, J ~ 13,. RECEIVE decoding: anneal on Vesuvius 503 spins gauge checking ~ σ 1, ~ σ 2, channel crossover probability, p Pr ( J~ AB J AB ) 1 1 p T Nishimori = log 2 p 500 bits received data

39 Gauge Symmetry (i) Decoding using the D-Wave Machine Transmit all zeros Degenerate decoded groundstates flip all data bits Unique decoded information ? (a) Single-shot decoding: random sample from distribution at T eff (b) Multiple run decoding: find groundstate with high probability Always map into the antiferromagnetic gauge on chip 1000 words per noise level; 1000 samples per word

40 Gauge Symmetry (ii) Real World decoding application Transmit Degenerate decoded groundstates Use redundant gauge check bits Transmit Degenerate decoded groundstates flip all data bits Unique decoded information ?

41 D-Wave Vesuvius Groundstate Decoding Gauge checking on all N data bits 1 1 decoder bit-error-rate No. of bits Rate of code channel crossover probability Shannon Limit

42 D-Wave Vesuvius Groundstate Decoding Gauge checking on all N data bits decoder bit-error-rate Shannon Limit channel crossover probability Phase transition near p = 0.2..

43 .. cf Katzgraber et al. PRX (2014) Phase diagram of spin glass on Chimera graph

44 D-Wave Vesuvius Groundstate Decoding Gauge checking on three redundant bits decoder bit-error-rate No. of bits Rate of code Shannon Limit channel crossover probability

45 D-Wave Vesuvius Finite-Temperature Decoding Gauge checking on three redundant bits decoder bit-error-rate Shannon Limit channel crossover probability x o finite-temp decoding ground-state decoding

46 p 0.15: T = 0 decoding is better than T > 0 decoding Otsubo et al. PRE (2012) T DW1 Overlap = 1 2 x BER T Nishimori T DW2 0 Γ DW1 Γ DW2

47 Conclusions Quantum mechanics plays a role up to N = 40 Lengthscale of entanglement? (needs a bipartite DW problem which MPS can handle) Ising code decoding: certainly not commercially competitive a useful tool for studying role of quantum fluctuations

48 Acknowledgements Walter Vinci Nick Chancellor Tanja Duric Phil Crowley Simone Severini Andrew Green Gabriel Aeppli Tameem Albash Anurag Mishra Daniel Lidar