Quantum vs Classical Optimization: A status update on the arms race. Helmut G. Katzgraber

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2 Quantum vs Classical Optimization: A status update on the arms race Helmut G. Katzgraber

3 Quantum vs Classical Optimization: A status update on the arms race Helmut G. Katzgraber

4 Quantum vs Classical Optimization: A status update on the arms race ~ =0 ~ > Helmut G. Katzgraber

5 Outline Some questions we would like answers to Current status of quantum vs classical optimization? What about quantum approaches for machine learning? If QA fails to deliver, can we still benefit? Think quantum inspired Texas A&M team: as well as S. F. C.

6 Outline Some questions we would like answers to Current status of quantum vs classical optimization? What about quantum approaches for machine learning? If QA fails to deliver, can we still benefit? Think quantum inspired Texas A&M team: as well as S. F. C.

7 Outline Some questions we would like answers to Current status of quantum vs classical optimization? What about quantum approaches for machine learning? If QA fails to deliver, can we still benefit? Think quantum inspired Texas A&M team: C. Pattison missing C. Fang H. Munoz-B. J. Chancellor Dr. W. Wang Dr. Z. Zhu A. Barzegar A. Ochoa as well as S. F. C.

8 Why quantum annealing? Optimization! Selected problems of interest: Constraint satisfaction (SAT) Number partitioning Minimum vertex covers Traveling salesman problem, (x11orx12)and(x21orx22)... min vertex cover NPP What do all these have in common? Rough cost function landscapes. They are problems in NP (also typical hard). All map onto Quadratic Unconstrained Binary Optimization (QUBO) problems. NX H(S i )= Q ij S i S j S i 2 {±1} i6=j NP Problems NP-complete P Problems

9 Why quantum annealing? Optimization! Selected problems of interest: Constraint satisfaction (SAT) Number partitioning Minimum vertex covers Traveling salesman problem, (x11orx12)and(x21orx22)... min vertex cover NPP What do all these have in common? Rough cost function landscapes. They are problems in NP (also typical hard). All map onto Quadratic Unconstrained Binary Optimization (QUBO) problems. NX H(S i )= Q ij S i S j S i 2 {±1} i6=j NP Problems NP-complete P Problems

10 Why quantum annealing? Optimization! Selected problems of interest: Constraint satisfaction (SAT) Number partitioning Minimum vertex covers Traveling salesman problem, (x11orx12)and(x21orx22)... min vertex cover NPP What do all these have in common? Rough cost function landscapes. They are problems in NP (also typical hard). All map onto Quadratic Unconstrained Binary Optimization (QUBO) problems. NX H(S i )= Q ij S i S j S i 2 {±1} min S i i6=j NP Problems NP-complete P Problems

11 Why quantum annealing? Optimization! Selected problems of interest: Constraint satisfaction (SAT) Number partitioning Minimum vertex covers Traveling salesman problem, (x11orx12)and(x21orx22)... min vertex cover NPP What do all these have in common? Rough cost function landscapes. They are problems in NP (also typical hard). All map onto Quadratic Unconstrained Binary Optimization (QUBO) problems. NX H(S i )= Q ij S i S j S i 2 {±1} min S i i6=j NP Problems NP-complete P Problems Good QUBO solvers & fast architectures needed!

12 Moore s Law is comingnewstofeature an end end of Moore s law: Four possible ways to overcome themoore S LORE Build larger silicon-based computers. Develop faster silicon-based technologies. Focus on faster algorithms. Go beyond standard silicon architectures. G. Moore (1965) 10 Here, deep synergy between For the past five decades, the number of transistors per microprocessor chip a rough measure of processing power has doubled about every two years, in step with Moore s law (top). Chips also increased their clock speed, or rate of executing instructions, until 2004, when speeds were capped to limit heat. As computers increase in power and shrink in size, a new class of machines has emerged roughly every ten years (bottom) Physics, quantum information, and computer science. transistor count Transistors per chip clock speed (MHz) adapted from Nature (2016) Clock speeds (MHz) to sm o tu s e e r c th m sm w p b fi im e c th w

13 Moore s Law is comingnewstofeature an end end of Moore s law: Four possible ways to overcome themoore S LORE Build larger silicon-based computers. Develop faster silicon-based technologies. Focus on faster algorithms. Go beyond standard silicon architectures. G. Moore (1965) 10 Here, deep synergy between For the past five decades, the number of transistors per microprocessor chip a rough measure of processing power has doubled about every two years, in step with Moore s law (top). Chips also increased their clock speed, or rate of executing instructions, until 2004, when speeds were capped to limit heat. As computers increase in power and shrink in size, a new class of machines has emerged roughly every ten years (bottom) Physics, quantum information, and computer science. transistor count Transistors per chip clock speed (MHz) adapted from Nature (2016) Clock speeds (MHz) to sm o tu s e e r c th m sm w p b fi im e c th w

14 Moore s Law is comingnewstofeature an end 1010 Here, deep synergy between Physics, quantum information, and computer science. G. Moore (1965) 10 8 transistor count not scalable Four possible ways to overcome the end of Moore s law: MOORE S LORE Build larger silicon-based computers. For the past five decades, the number of transistors per microprocessor chip a rough measure of processing power has doubled about every Develop faster silicon-based technologies. two years, in step with Moore s law (top). Chips also increased their clock speed, or rate of executing instructions, until 2004, when speeds were Focus on faster algorithms. capped to limit heat. As computers increase in power and shrink in size, a new class of machines has emerged roughly every ten years (bottom). Go beyond standard silicon architectures Transistors per chip clock speed (MHz) adapted from Nature (2016) Clock speeds (MHz) to sm o tu s e e r c th m sm w p b fi im e c th w

15 Moore s Law is comingnewstofeature an end 1010 Here, deep synergy between Physics, quantum information, and computer science. G. Moore (1965) 10 8 transistor count not scalable Four possible ways to overcome the end of Moore s law: MOORE S LORE Build larger silicon-based computers. already close to fab limits For the past five decades, the number of transistors per microprocessor chip a rough measure of processing power has doubled about every Develop faster silicon-based technologies. two years, in step with Moore s law (top). Chips also increased their clock speed, or rate of executing instructions, until 2004, when speeds were Focus on faster algorithms. capped to limit heat. As computers increase in power and shrink in size, a new class of machines has emerged roughly every ten years (bottom). Go beyond standard silicon architectures Transistors per chip clock speed (MHz) adapted from Nature (2016) Clock speeds (MHz) to sm o tu s e e r c th m sm w p b fi im e c th w

16 Moore s Law is comingnewstofeature an end 1010 Here, deep synergy between Physics, quantum information, and computer science. G. Moore (1965) 10 8 transistor count not scalable Four possible ways to overcome the end of Moore s law: MOORE S LORE Build larger silicon-based computers. already close to fab limits For the past five decades, the number of transistors per microprocessor chip a rough measure of processing power has doubled about every Develop faster silicon-based technologies. two years, in step with Moore s law (top). Chips also increased their clock potentially disruptive speed, or rate of executing instructions, until 2004, when speeds were Focus on faster algorithms. capped to limit heat. As computers increase in power and shrink in size, a new class of machines has emerged roughly every ten years (bottom). Go beyond standard silicon architectures Transistors per chip clock speed (MHz) adapted from Nature (2016) Clock speeds (MHz) to sm o tu s e e r c th m sm w p b fi im e c th w

17 Moore s Law is comingnewstofeature an end 1010 Here, deep synergy between Physics, quantum information, and computer science. G. Moore (1965) 10 8 transistor count not scalable Four possible ways to overcome the end of Moore s law: MOORE S LORE Build larger silicon-based computers. already close to fab limits For the past five decades, the number of transistors per microprocessor chip a rough measure of processing power has doubled about every Develop faster silicon-based technologies. two years, in step with Moore s law (top). Chips also increased their clock potentially disruptive speed, or rate of executing instructions, until 2004, when speeds were Focus on faster algorithms. capped to limit heat. As computers increase in power and shrink in size, a new class of machines has emerged roughly every ten years (bottom). Go beyond standard silicon architectures Transistors per chip clock speed (MHz) adapted from Nature (2016) Clock speeds (MHz) to sm o tu s e e r c th m sm w p b fi im e c th w

18 Current state of the art: Special-purpose analog quantum annealers Antikythera ~ 80BC

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20 What is it? Semi-programmable analog annealer superconducting flux qubits. Controversial performance. Still, huge technological feat What can it do? It can minimize QUBOs post embedding onto the machine s hardwired Chimera topology. Limitations: Low connectivity. Analog noise.

21 What is it? Semi-programmable analog annealer superconducting flux qubits. Controversial performance. Still, huge technological feat What can it do? It can minimize QUBOs post embedding onto the machine s hardwired Chimera topology. Limitations: Low connectivity. Analog noise.

22 K44 What is it? Semi-programmable analog annealer superconducting flux qubits. Controversial performance. Still, huge technological feat What can it do? It can minimize QUBOs post embedding onto the machine s hardwired Chimera topology. Limitations: Low connectivity. Analog noise.

23 How do quantum annealers optimize?

24 How do quantum annealers optimize? Sequentially.

25 Classical Analog: Simulated Annealing (SA) Annealing: 7000 year-old neolithic technology. Slowly cool to remove imperfections. Kirkpatrick et al., Science (83) German copper axe Simulated Annealing (SA): Stochastically sample using Monte Carlo. H({S}) If the system is thermalized, cool it. The slower the cooling, the better, e.g., Geman & Geman T (t) =a bt Problem: SA is inefficient for complex systems. Solution: Multiple restarts & statistics gathering.

26 Classical Analog: Simulated Annealing (SA) Annealing: 7000 year-old neolithic technology. Slowly cool to remove imperfections. Kirkpatrick et al., Science (83) German copper axe Simulated Annealing (SA): Stochastically sample using Monte Carlo. H({S}) If the system is thermalized, cool it. The slower the cooling, the better, e.g., Geman & Geman T (t) =a bt Problem: SA is inefficient for complex systems. Solution: Multiple restarts & statistics gathering.

27 Classical Analog: Simulated Annealing (SA) Annealing: 7000 year-old neolithic technology. Slowly cool to remove imperfections. Simulated Annealing (SA): Stochastically sample using Monte Carlo. H({S}) If the system is thermalized, cool it. The slower the cooling, the better, e.g., Kirkpatrick et al., Science (83) Geman & Geman T (t) =a bt Problem: SA is inefficient for complex systems. Solution: Multiple restarts & statistics gathering.

28 Classical Analog: Simulated Annealing (SA) Annealing: 7000 year-old neolithic technology. Slowly cool to remove imperfections. Simulated Annealing (SA): Stochastically sample using Monte Carlo. H({S}) If the system is thermalized, cool it. The slower the cooling, the better, e.g., Kirkpatrick et al., Science (83) Geman & Geman T (t) =a bt Problem: SA is inefficient for complex systems. Solution: Multiple restarts & statistics gathering.

29 Quantum Annealing (QA) Idea: Use quantum fluctuations instead of thermal. Sequential algorithm like SA. Theoretical advantages over SA: Fluctuations determine the tunneling radius. Not limited to a local search. Implementation in DW device (transverse-field QA): Apply a transverse field that does not commute: H(S i )= NX i6=j Q ij S i S j H(S i )= Morita & Nishimori (06) Kadowaki & Nishimori (98) Farhi et al. (00) Reduce the fluctuation amplitude D via a given annealing protocol. NX i6=j [S x,s z ] 6= 0 Q ij S z i S z i D NX i S x i

30 Quantum Annealing (QA) Idea: Use quantum fluctuations instead of thermal. Sequential algorithm like SA. Theoretical advantages over SA: Fluctuations determine the tunneling radius. Not limited to a local search. Implementation in DW device (transverse-field QA): Apply a transverse field that does not commute: H(S i )= NX i6=j Q ij S i S j H(S i )= Morita & Nishimori (06) Kadowaki & Nishimori (98) Farhi et al. (00) Reduce the fluctuation amplitude D via a given annealing protocol. NX i6=j [S x,s z ] 6= 0 Q ij S z i S z i D NX i S x i

31 Promising signs of quantum speedup? Denchev et al. (Dec. 2015) see Mandrà, Zhu, Perdomo-O. & Katzgraber (PRA, arxiv: )

32 Google s 10 8 results slope vs offset QMC and SA single-core annealing time (µs) TTS in µs th 75th 50th QMC SA D-Wave Denchev et al. (15) D-Wave annealing time (µs) DW2 annealing time in Problem size (bits) N [problem size]

33 Google s 10 8 results slope vs offset QMC and SA single-core annealing time (µs) TTS in µs th 75th 50th QMC SA D-Wave Denchev et al. (15) D-Wave annealing time (µs) DW2 annealing time in Problem size (bits) N [problem size]

34 Google s 10 8 results slope vs offset QMC and SA single-core annealing time (µs) TTS in µs (b) 85th 75th Q J = 1 50th Q J =+1 < 0.5 h1 = 1 QMC SA D-Wave h2 = h1 Denchev et al. (15) (a) D-Wave annealing time (µs) DW2 annealing time in Problem size (bits) N [problem size]

35 th (b) 75th 50th Denchev et al. (15)proba J = 11 Q J= Q = < (a) 106 h1 SA h2 = QMC h1 = TTS inannealing µs QMC and SA single-core time (µs) results slope vs offset D-Wave 2 N X X hi Si 900 H(S Qij S i ) = i Sj 700 Figure 1: Sketch of the weak-strong clusters and networks. Problem size (bits) i K j Ni6= [problem size]two (a) Structure of a weak-strong cluster. 4,4 cells of the corre is the closest The c the DW2X. T N is t approximate date [15] and con th despiteinboth MonteofCarlo Tw the curves As exi Carlo and th tiona first solid ev core.t capabilities possess, it is Wave annealing time (µs) DW2D-Wave annealing time in µs Google s 8 10 parison to a methods. W the results o any sequenti using of outperfor Th methods. We as aof c structure timiz ison. Howev

36 Denchev et al. (15)proba spin-glass 85th backbone (b) th 50th J = 11 Q J= Q = < (a) 106 h1 SA h2 = QMC h1 = TTS inannealing µs QMC and SA single-core time (µs) results slope vs offset D-Wave 2 N X X hi Si 900 H(S Qij S i ) = i Sj 700 Figure 1: Sketch of the weak-strong clusters and networks. Problem size (bits) i K j Ni6= [problem size]two (a) Structure of a weak-strong cluster. 4,4 cells of the corre is the closest The c the DW2X. T N is t approximate date [15] and con th despiteinboth MonteofCarlo Tw the curves As exi Carlo and th tiona first solid ev core.t capabilities possess, it is Wave annealing time (µs) DW2D-Wave annealing time in µs Google s 8 10 parison to a methods. W the results o any sequenti using of outperfor Th methods. We as aof c structure timiz ison. Howev

37 Google s 10 8 results slope vs offset QMC and SA single-core annealing time (µs) TTS in µs th 75th 50th QMC SA D-Wave Denchev et al. (15) D-Wave annealing time (µs) DW2 annealing time in Problem size (bits) N [problem size]

38 Google s 10 8 results slope vs offset QMC and SA single-core annealing time (µs) TTS in µs th 75th 50th Catapult + QMC QMC SA D-Wave Denchev et al. (15) D-Wave annealing time (µs) DW2 annealing time in Problem size (bits) N [problem size]

39 Google s 10 8 results slope vs offset QMC and SA single-core annealing time (µs) TTS in µs th 75th 50th Catapult + QMC QMC SA D-Wave Denchev et al. (15) D-Wave annealing time (µs) DW2 annealing time in Problem size (bits) N [problem size] Better scaling of DW and quantum inspired.

40 ~ =0 ~ > 0 0 0

41 ~ =0 ~ > 0 0 1

42 What if we use better algorithms? Tailored to the problems and/or underlying graph: Hamze-de Freitas-Selby algorithm (HFS). Hamze et al. (12) Hybrid cluster methods (HCM). Super-spin approximation (SS). Not tailored to the problems and/or underlying graph: Parallel tempering & isoenergetic cluster Population annealing (particle swarm) sequential Monte Carlo (PA). optimizer (PT+ICM). Venturelli, et al. (15) Zhu (16) Zhu, Ochoa, Katzgraber PRL (15) Reminder Sequential methods used in the Google study: Simulated annealing (SA). Kirkpatrick et al. (83) Quantum Monte Carlo (QMC). Denchev et al. (15) D-Wave 2X (DW2). Wang et al., PRE (15)

43 What if we use better algorithms? Tailored to the problems and/or underlying graph: Hamze-de Freitas-Selby algorithm (HFS). Hamze et al. (12) Hybrid cluster methods (HCM). Super-spin approximation (SS). Not tailored to the problems and/or underlying graph: Parallel tempering & isoenergetic cluster Population annealing (particle swarm) sequential Monte Carlo (PA). optimizer (PT+ICM). Venturelli, et al. (15) Zhu (16) Zhu, Ochoa, Katzgraber PRL (15) Reminder Sequential methods used in the Google study: Simulated annealing (SA). Kirkpatrick et al. (83) Quantum Monte Carlo (QMC). Denchev et al. (15) D-Wave 2X (DW2). Wang et al., PRE (15)

44 What if we use better algorithms? Tailored to the problems and/or underlying graph: Hamze-de Freitas-Selby algorithm (HFS). Hamze et al. (12) Hybrid cluster methods (HCM). Super-spin approximation (SS). Not tailored to the problems and/or underlying graph: Parallel tempering & isoenergetic cluster Population annealing (particle swarm) sequential Monte Carlo (PA). optimizer (PT+ICM). Venturelli, et al. (15) Zhu (16) Zhu, Ochoa, Katzgraber PRL (15) MaxSAT 2016 winner Reminder Sequential methods used in the Google study: Simulated annealing (SA). Kirkpatrick et al. (83) Quantum Monte Carlo (QMC). Denchev et al. (15) D-Wave 2X (DW2). Wang et al., PRE (15)

45 What if we use better algorithms? Tailored to the problems and/or underlying graph: Hamze-de Freitas-Selby algorithm (HFS). Hamze et al. (12) Hybrid cluster methods (HCM). Super-spin approximation (SS). Not tailored to the problems and/or underlying graph: Parallel tempering & isoenergetic cluster Population annealing (particle swarm) sequential Monte Carlo (PA). optimizer (PT+ICM). Venturelli, et al. (15) Zhu (16) Zhu, Ochoa, Katzgraber PRL (15) MaxSAT 2016 winner Reminder Sequential methods used in the Google study: Simulated annealing (SA). Kirkpatrick et al. (83) Quantum Monte Carlo (QMC). Denchev et al. (15) D-Wave 2X (DW2). Wang et al., PRE (15) Catapult?

46 Asymptotic scaling exponent b (slope) b [50%] b [50%, main scaling exponent] (a + b n + c log 10 ( n)) fit T poly( p n)10 a+bp n (a + b n) fit SS HFS PT+ICM RMC+ICM HCM QMC DW2 SA PA

47 Asymptotic scaling exponent b (slope) b [50%] b [50%, main scaling exponent] (a + b n + c log 10 ( n)) fit T poly( p n)10 a+bp n (a + b n) fit SS HFS PT+ICM RMC+ICM HCM QMC DW2 smaller means better scaling SA PA

48 Asymptotic scaling exponent b (slope) b [50%] b [50%, main scaling exponent] sequential (a + b n + c log 10 ( n)) fit T poly( p n)10 a+bp n tailored (a + b n) fit SS HFS PT+ICM RMC+ICM HCM QMC DW2 not tailored tailored smaller means better scaling SA PA

49 Asymptotic scaling exponent b (slope) b [50%] b [50%, main scaling exponent] sequential tailored not tailored tailored smaller means better scaling SA PA HCM QMC DW2 (a + b n + c log 10 ( n)) fit T poly( p n)10 a+bp n (a + b n) fit SS HFS PT+ICM RMC+ICM Only sequential quantum speedup.

50 ~ =0 ~ > 0 0 1

51 ~ =0 ~ > 0 1 1

52 Most recent D-Wave benchmarks King et al. (17) see Mandrà, Katzgraber & Thomas (QST, arxiv: )

53 D-Wave s frustrated cluster loop problems α = 0.80, ρ = 5 King et al. (17) MWPM (no broken qubits) MWPM DW2000Q, TTS 1 DW2000Q, TTS 2 ICM (logical), TTS 2 TTS [µs] QMC SA DW2000Q p n [number of logical variables]

54 D-Wave s frustrated cluster loop problems α = 0.80, ρ = 5 King et al. (17) MWPM (no broken qubits) MWPM DW2000Q, TTS 1 DW2000Q, TTS 2 ICM (logical), TTS 2 TTS [µs] Catapult QMC + QMC SA DW2000Q p n [number of logical variables]

55 D-Wave s frustrated cluster loop problems TTS [µs] α = 0.80, ρ = 5 Ruggedness of FCLs (spin-glass backbone) fools codes. The logical problem is defined on K 44 cells and is therefore planar. Catapult QMC + QMC SA DW2000Q MWPM (no broken qubits) MWPM DW2000Q, TTS 1 DW2000Q, TTS 2 ICM (logical), TTS p n [number of logical variables] King et al. (17)

56 D-Wave s frustrated cluster loop problems TTS [µs] α = 0.80, ρ = 5 Ruggedness of FCLs (spin-glass backbone) fools codes. The logical problem is defined on K 44 cells and is therefore planar. Catapult QMC + QMC SA DW2000Q MWPM (no broken qubits) MWPM DW2000Q, TTS 1 DW2000Q, TTS 2 ICM (logical), TTS p n [number of logical variables] King et al. (17) Why is this a problem? Planar problems are polynomial (P class). Exact algorithms exist.

57 Using minimum-weight perfect matching King et al. (17) Edmonds (61) Mandrà et al. (17) TTS [µs] mwpm (fully-chimera) mwpm DW2kQ, 1/p tts DW2kQ, log(0.01)/log(1-p) tts p n [number of logical variables]

58 Using minimum-weight perfect matching King et al. (17) DW2000Q Edmonds (61) Mandrà et al. (17) MWPM TTS [µs] mwpm (fully-chimera) mwpm DW2kQ, 1/p tts DW2kQ, log(0.01)/log(1-p) tts p n [number of logical variables]

59 Using minimum-weight perfect matching King et al. (17) DW2000Q Edmonds (61) Mandrà et al. (17) MWPM TTS [µs] mwpm (fully-chimera) mwpm DW2kQ, 1/p tts DW2kQ, log(0.01)/log(1-p) tts Exponentially faster 1000 p n [number of logical variables] than DW2000Q

60 Using minimum-weight perfect matching King et al. (17) DW2000Q Edmonds (61) Mandrà et al. (17) MWPM TTS [µs] mwpm (fully-chimera) mwpm DW2kQ, 1/p tts DW2kQ, log(0.01)/log(1-p) tts Exponentially faster 1000 p n [number of logical variables] than DW2000Q

61 ~ =0 ~ > 0 1 1

62 ~ =0 ~ > 0 2 1

63 Fair sampling A key ingredient in ML A see also Mandrà, Zhu & Katzgraber (PRL, arxiv: )

64 What is fair sampling? Definition (fair sampling): Ability of an algorithm to find uncorrelated solutions to a problem with (almost) the same probability. Why is this important? Sometimes solutions are more important than the optimum (SAT filters, #SAT, machine learning, ). Some solutions might be more convenient due to additional constraints. Algorithm benchmarking: Standard Stringent Find the optimum fast and reliably. Find all minimizing configurations equiprobably.

65 What is fair sampling? Definition (fair sampling): Ability of an algorithm to find uncorrelated solutions to a problem with (almost) the same probability. Why is this important? Sometimes solutions are more important than the optimum (SAT filters, #SAT, machine learning, ). Some solutions might be more convenient due to additional constraints. Algorithm benchmarking: Standard Stringent Find the optimum fast and reliably. Find all minimizing configurations equiprobably.

66 What is fair sampling? Definition (fair sampling): Ability of an algorithm to find uncorrelated solutions to a problem with (almost) the same probability. Why is this important? Sometimes solutions are more important than the optimum (SAT filters, #SAT, machine learning, ). Some solutions might be more convenient due to additional constraints. Algorithm benchmarking: Standard Stringent current state of the art is PT+ICM Find the optimum fast and reliably. Find all minimizing configurations equiprobably.

67 Can transverse-field QA sample fairly? 5-variable toy model suggests bias: 1i 2i 3i 1 J ij =+1 J ij = 1 H= X hijij ij S i S j Matsuda, Nishimori, Katzgraber (NJP 2009) What about quantum annealers? annealing time Design problems with known degeneracy: H= X ij S i S j J ij 2 {±5, ±6, ±7} hijij N GS =3 2 k,k 2 N P [probability of states] Study the distribution of ground states for fixed NGS. log(hits) fair sampling rankgs

68 Can transverse-field QA sample fairly? 5-variable toy model suggests bias: 2i 3i J ij =+1 J ij = 1 H= X hijij ij S i S j Matsuda, Nishimori, Katzgraber (NJP 2009) What about quantum annealers? annealing time Design problems with known degeneracy: H= X ij S i S j J ij 2 {±5, ±6, ±7} hijij N GS =3 2 k,k 2 N P [probability of states] Study the distribution of ground states for fixed NGS. log(hits) fair sampling rankgs

69 Can transverse-field QA sample fairly? 5-variable toy model suggests bias: 2i 3i J ij =+1 J ij = 1 H= X hijij ij S i S j Matsuda, Nishimori, Katzgraber (NJP 2009) What about quantum annealers? annealing time Design problems with known degeneracy: H= X ij S i S j J ij 2 {±5, ±6, ±7} hijij N GS =3 2 k,k 2 N P [probability of states] Study the distribution of ground states for fixed NGS. log(hits) fair sampling rankgs

70 Can transverse-field QA sample fairly? 5-variable toy model suggests bias: 2i 3i J ij =+1 J ij = 1 H= X hijij ij S i S j Matsuda, Nishimori, Katzgraber (NJP 2009) What about quantum annealers? annealing time Design problems with known degeneracy: H= X ij S i S j J ij 2 {±5, ±6, ±7} hijij N GS =3 2 k,k 2 N P [probability of states] Study the distribution of ground states for fixed NGS. log(hits) rankgs unfair

71 Transverse-field QA is exponentially biased Number of Hits [Average] Number of hits (averaged over samples) sample data for N = RankGS/(32 k ) Rank gs /(3 2 k ) c = 9, k = 2 c = 9, k = 3 c = 9, k = 4 c = 9, k = 5 k = 2 (NGS = 12) k = 3 (NGS = 24) k = 4 (NGS = 48) k = 5 (NGS = 96)

72 Transverse-field QA is exponentially biased Number of Hits [Average] Number of hits (averaged over samples) sample data for N = RankGS/(32 k ) Rank gs /(3 2 k ) c = 9, k = 2 c = 9, k = 3 c = 9, k = 4 c = 9, k = 5 k = 2 (NGS = 12) k = 3 (NGS = 24) k = 4 (NGS = 48) k = 5 (NGS = 96) Standard QA will need tweaks for fair sampling.

73 ~ =0 ~ > 0 2 1

74 ~ =0 ~ > 0 3 1

75 ~ =0 ~ > 0 3 analog 1 QA

76 ~ =0 ~ > analog QA Look out for IARPA s QEO report on QA.

77 ~ =0 ~ > analog QA Look out for IARPA s QEO report on QA. However Soon superseded by digital?

78 Quantum vs Classical Optimization: A status update on the arms race Classical optimization pushes quantum technology. Quantum developments leverage classical quantum inspired methods. ML could benefit from quantum samplers if these can sample fairly. To date, no application speedup or better scaling of quantum annealing. contact-us@intractable.lol

79 Quantum vs Classical Optimization: A status update on the arms race Thank you. Classical optimization pushes quantum technology. Quantum developments leverage classical quantum inspired methods. ML could benefit from quantum samplers if these can sample fairly. To date, no application speedup or better scaling of quantum annealing. contact-us@intractable.lol

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