Polynomial-time classical simulation of quantum ferromagnets

Transcription

1 Polynomial-time classical simulation of quantum ferromagnets Sergey Bravyi David Gosset IBM PRL 119, (2017) arxiv:

2 Quantum Monte Carlo: a powerful suite of probabilistic classical simulation algorithms for quantum many-body systems. Can simulate systems orders of magnitude larger than with exact diagonalization

3 [Sandvik, Hamer 1999] Ground state properties of 2D ferromagnetic XY model on L L grid. H = <ij> X i X j + Y i Y j They also perform finite-temperature numerics up to L = 64.

4 [Sandvik, Hamer 1999] Ground state properties of 2D ferromagnetic XY model on L L grid. H = <ij> X i X j + Y i Y j They also perform finite-temperature numerics up to L = 64. This is a classical simulation of up to 4096 spins! What s the catch? How does it work?

5 What s the catch? Quantum Monte Carlo can only be used to study stoquastic Hamiltonians x H x R y H x 0 x y Stoquastic i.e., sign-problem free

6 What s the catch? Quantum Monte Carlo can only be used to study stoquastic Hamiltonians x H x R y H x 0 x y Stoquastic i.e., sign-problem free Examples: Particle in a potential H = p2 2m + V( Ԧx)

7 What s the catch? Quantum Monte Carlo can only be used to study stoquastic Hamiltonians x H x R y H x 0 x y Stoquastic i.e., sign-problem free Examples: Particle in a potential H = p2 2m + V( Ԧx) Hopping and interacting bosons H = (a i a j + a j a i ) + V( n) <ij>

8 What s the catch? Quantum Monte Carlo can only be used to study stoquastic Hamiltonians x H x R y H x 0 x y Stoquastic i.e., sign-problem free Examples: Particle in a potential H = p2 2m + V( Ԧx) Hopping and interacting bosons H = (a i a j + a j a i ) + V( n) <ij> Quantum annealing Hamiltonians H = (1 s) X i + sv i ԦZ

9 How does it work? QMC is based on a probabilistic representation of the Gibbs state ρ = e βh Z(β) Z β = Tr(e βh ) A collection of samples from a certain probability distribution associated with ρ are sufficient to evaluate expectation values of observables.

10 How does it work? Z β = Tr(e βh ) Partition function at temperature T = β 1

11 How does it work? Z β = Tr(e βh ) Partition function at temperature T = β 1 = Tr((e ΔH ) M ) Δ 1 M = βδ 1

12 How does it work? Z β = Tr(e βh ) Partition function at temperature T = β 1 = Tr((e ΔH ) M ) Δ 1 M = βδ 1 z 1 I ΔH z 2 z 2 I ΔH z 3 z M I ΔH z 1 z 1,z 2, z M {0,1} n Insert complete sets of states

13 How does it work? Z β = Tr(e βh ) Partition function at temperature T = β 1 = Tr((e ΔH ) M ) Δ 1 M = βδ 1 z 1 I ΔH z 2 z 2 I ΔH z 3 z M I ΔH z 1 z 1,z 2, z M {0,1} n Insert complete sets of states p z 1, z 2, z M 0 (use stoquasticity)

14 How does it work? Z β = Tr(e βh ) Partition function at temperature T = β 1 = Tr((e ΔH ) M ) Δ 1 M = βδ 1 z 1 I ΔH z 2 z 2 I ΔH z 3 z M I ΔH z 1 z 1,z 2, z M {0,1} n Insert complete sets of states p z 1, z 2, z M 0 (use stoquasticity) QMC uses a classical Markov chain to equilibrate to the probability distribution proportional to p(z 1, z 2,, z M ).

15 How does it work? Z β = Tr(e βh ) Partition function at temperature T = β 1 = Tr((e ΔH ) M ) Δ 1 M = βδ 1 z 1 I ΔH z 2 z 2 I ΔH z 3 z M I ΔH z 1 z 1,z 2, z M {0,1} n Insert complete sets of states p z 1, z 2, z M 0 (use stoquasticity) QMC uses a classical Markov chain to equilibrate to the probability distribution proportional to p(z 1, z 2,, z M ). Physical properties are computed as expectation values.

16 In additional to empirical evidence, there is also complexity-theoretic evidence suggesting that stoquastic Hamiltonians may be easier to simulate. Local Hamiltonian problem: Given a local Hamiltonian H and two numbers a < b, decide if the ground energy of H is a or b. (promised that one case holds).

17 In additional to empirical evidence, there is also complexity-theoretic evidence suggesting that stoquastic Hamiltonians may be easier to simulate. Local Hamiltonian problem: Given a local Hamiltonian H and two numbers a < b, decide if the ground energy of H is a or b. (promised that one case holds). The local Hamiltonian problem is QMA-complete in general. [Kitaev 99]

18 In additional to empirical evidence, there is also complexity-theoretic evidence suggesting that stoquastic Hamiltonians may be easier to simulate. Local Hamiltonian problem: Given a local Hamiltonian H and two numbers a < b, decide if the ground energy of H is a or b. (promised that one case holds). The local Hamiltonian problem is QMA-complete in general. [Kitaev 99] For stoquastic local Hamiltonians it is StoqMA-complete (MA StoqMA AM) [Bravyi, Divincenzo, Oliveira, Terhal 2006]

19 In additional to empirical evidence, there is also complexity-theoretic evidence suggesting that stoquastic Hamiltonians may be easier to simulate. Local Hamiltonian problem: Given a local Hamiltonian H and two numbers a < b, decide if the ground energy of H is a or b. (promised that one case holds). The local Hamiltonian problem is QMA-complete in general. [Kitaev 99] For stoquastic local Hamiltonians it is StoqMA-complete (MA StoqMA AM) [Bravyi, Divincenzo, Oliveira, Terhal 2006] For classical local Hamiltonians it is NP-complete.

20 Examples: H = h(i, j) 1 i<j n LH problem Ising model h i, j = α ij Z i Z j NP-complete (Classical) Transverse-field Ising model h i, j = α ij X i X j γ i Z i γ j Z j StoqMA-complete [Bravyi, Hastings 2014] (Stoquastic) XY model h i, j = α ij (X i X j + Y i Y j ) QMA-complete [Cubitt Montanaro 2013] (Quantum)

21 Examples: H = h(i, j) 1 i<j n LH problem Ising model h i, j = α ij Z i Z j NP-complete (Classical) Transverse-field Ising model h i, j = α ij X i X j γ i Z i γ j Z j StoqMA-complete [Bravyi, Hastings 2014] (Stoquastic) XY model h i, j = α ij (X i X j + Y i Y j ) QMA-complete [Cubitt Montanaro 2013] (Quantum) These examples illustrate three flavours of intractable constraint satisfaction problems. (they represent all nontrivial possibilities within the framework of [Cubitt Montanaro 2013])

22 Can we identify cases where Quantum Monte Carlo is provably efficient? H = 1 i<j n h(i, j) Ferromagnetic Ising model h i, j = α ij Z i Z j Approximate Ground energy Trivial Approximate Partition Function

23 Can we identify cases where Quantum Monte Carlo is provably efficient? H = 1 i<j n h(i, j) Approximate Ground energy Approximate Partition Function Ferromagnetic Ising model h i, j = α ij Z i Z j Trivial In BPP [Jerrum,Sinclair 1989]

24 Can we identify cases where Quantum Monte Carlo is provably efficient? H = 1 i<j n h(i, j) Approximate Ground energy Approximate Partition Function Ferromagnetic Ising model h i, j = α ij Z i Z j Trivial In BPP [Jerrum,Sinclair 1989] Ferromagnetic Transverse-field Ising model h i, j = α ij X i X j γ i Z i γ j Z j

25 Can we identify cases where Quantum Monte Carlo is provably efficient? H = 1 i<j n h(i, j) Approximate Ground energy Approximate Partition Function Ferromagnetic Ising model h i, j = α ij Z i Z j Trivial In BPP [Jerrum,Sinclair 1989] Ferromagnetic Transverse-field Ising model h i, j = α ij X i X j γ i Z i γ j Z j In BPP [Bravyi 2015] In BPP [Bravyi 2015]

26 Can we identify cases where Quantum Monte Carlo is provably efficient? H = 1 i<j n h(i, j) Approximate Ground energy Approximate Partition Function Ferromagnetic Ising model h i, j = α ij Z i Z j Trivial In BPP [Jerrum,Sinclair 1989] Ferromagnetic Transverse-field Ising model h i, j = α ij X i X j γ i Z i γ j Z j In BPP [Bravyi 2015] In BPP [Bravyi 2015] Ferromagnetic XY model h i, j = α ij (X i X j + Y i Y j )

27 Can we identify cases where Quantum Monte Carlo is provably efficient? H = 1 i<j n h(i, j) Approximate Ground energy Approximate Partition Function Ferromagnetic Ising model h i, j = α ij Z i Z j Trivial In BPP [Jerrum,Sinclair 1989] Ferromagnetic Transverse-field Ising model h i, j = α ij X i X j γ i Z i γ j Z j In BPP [Bravyi 2015] In BPP [Bravyi 2015] Ferromagnetic XY model h i, j = α ij (X i X j + Y i Y j ) In BPP [This talk] In BPP [This talk]

28 Open question : Can QMC be used to efficiently simulate quantum adiabatic algorithms with stoquastic Hamiltonians? H = (1 s) X i + sv i ԦZ [Bravyi Terhal 2008] [Hastings Freedman 2013] [Crosson Harrow 2016] [Jarret Jordan Lackey 2016]

29 I. Results

30 The Hamiltonian We consider Hamiltonians of the form H = b ij X i X j + c ij Y i Y j + i<j i=1 n d i (I + Z i ) Coefficients must satisfy b ij, c ij, d i 1 (sets energy scale) c ij b ij (ensures stoquasticity)

31 The Hamiltonian We consider Hamiltonians of the form H = b ij X i X j + c ij Y i Y j + i<j i=1 n d i (I + Z i ) b ij c ij 0 0 c ij b ij 0 0 b ij c ij c ij b ij c ij b ij (ensures stoquasticity)

32 The Hamiltonian We consider Hamiltonians of the form H = b ij X i X j + c ij Y i Y j + i<j i=1 n d i (I + Z i ) p ij Y i Y j X i X j + q ij ( Y i Y j X i X j ) p ij, q ij 0

33 The Hamiltonian We consider Hamiltonians of the form H = b ij X i X j + c ij Y i Y j + i<j i=1 n d i (I + Z i ) Special cases: d i = 0 c ij = 0 Classical Ferromagnetic Ising model c ij = 0 Ferromagnetic transverse-field Ising model b ij = 1 c ij = 1 Ferromagnetic XY model b ij = 1 c ij = 1 (name?)

34 Approximating the partition function Definition (ε-approximation) Write a ε b iff e ε b a e ε b

35 Approximating the partition function Definition (ε-approximation) Write a ε b iff e ε b a e ε b An ε-approximation of Z(β) can be used to compute an estimate of the free energy F = 1 β log Z(β) to within additive error O(εβ 1 ) and an estimate of the ground energy to within additive error O (ε + n)β 1.

36 Polynomial-time approximation algorithm Theorem There exists a classical randomized algorithm which, given H, β, and a precision parameter ε (0,1) outputs an estimate satisfying Z ε Z β with high probability. The runtime of the algorithm is poly n, β, ε 1 As a corollary we obtain an efficient algorithm to approximate the free energy and the ground energy to a given additive error.

37 Polynomial-time approximation algorithm Theorem There exists a classical randomized algorithm which, given H, β, and a precision parameter ε (0,1) outputs an estimate satisfying Z ε Z β with high probability. The runtime of the algorithm is poly n, β, ε 1 = O n 115 (1 + β 46 ε 25 ). As a corollary we obtain an efficient algorithm to approximate the free energy and the ground energy to a given additive error.

38 Polynomial-time approximation algorithm Theorem There exists a classical randomized algorithm which, given H, β, and a precision parameter ε (0,1) outputs an estimate satisfying Z ε Z β with high probability. The runtime of the algorithm is poly n, β, ε 1 = O n 115 (1 + β 46 ε 25 ). As a corollary we obtain an efficient algorithm to approximate the free energy and the ground energy to a given additive error. The algorithm is not practical.

39 Polynomial-time approximation algorithm Theorem There exists a classical randomized algorithm which, given H, β, and a precision parameter ε (0,1) outputs an estimate satisfying Z ε Z β with high probability. The runtime of the algorithm is poly n, β, ε 1 = O n 115 (1 + β 46 ε 25 ). As a corollary we obtain an efficient algorithm to approximate the free energy and the ground energy to a given additive error. The algorithm is not practical. The proof is based on a reduction to counting perfect matchings

40 II. Perfect matchings

41 A perfect matching of a graph G = V, E is a subset of edges M E such that every vertex is incident to exactly one edge in M Example:

42 A perfect matching of a graph G = V, E is a subset of edges M E such that every vertex is incident to exactly one edge in M Example: The red edges are a perfect matching

43 A perfect matching of a graph G = V, E is a subset of edges M E such that every vertex is incident to exactly one edge in M Example: The red edges are a perfect matching

44 Now suppose the graph has edge weights w e e E. Each perfect matching M is assigned weight e M w e

45 Now suppose the graph has edge weights w e e E. Each perfect matching M is assigned weight Perfect matching sum: e M w e PerfMatch G = w e Perfect matchings M e M

46 Now suppose the graph has edge weights w e e E. Each perfect matching M is assigned weight Perfect matching sum: e M w e PerfMatch G = w e Perfect matchings M e M Example: a d b PerfMatch G = ac + bd c

47 A nearly perfect matching of a graph G = V, E is a subset of edges M E such that every vertex is incident to exactly one edge in M, except for 2 vertices which are untouched.

48 A nearly perfect matching of a graph G = V, E is a subset of edges M E such that every vertex is incident to exactly one edge in M, except for 2 vertices which are untouched. Nearly perfect matching sum: NearPerfMatch G = w e Nearly Perfect matchings M e M

49 Suppose G is a graph with nonnegative edge weights. Exactly compute PerfMatch(G) ε-approximation to PerfMatch(G) Planar graphs: In P Fisher, Kasteleyn, Temperley algorithm Bipartite graphs: (permanent of nonnegative matrix) General graphs:

50 Suppose G is a graph with nonnegative edge weights. Planar graphs: Exactly compute PerfMatch(G) In P Fisher, Kasteleyn, Temperley algorithm ε-approximation to PerfMatch(G) In P Bipartite graphs: (permanent of nonnegative matrix) General graphs:

51 Suppose G is a graph with nonnegative edge weights. Planar graphs: Exactly compute PerfMatch(G) In P Fisher, Kasteleyn, Temperley algorithm ε-approximation to PerfMatch(G) In P Bipartite graphs: (permanent of nonnegative matrix) #P-hard [Valiant 1979] In BPP [Jerrum, Sinclair, Vigoda 2004] General graphs:

52 Suppose G is a graph with nonnegative edge weights. Planar graphs: Exactly compute PerfMatch(G) In P Fisher, Kasteleyn, Temperley algorithm ε-approximation to PerfMatch(G) In P Bipartite graphs: (permanent of nonnegative matrix) #P-hard [Valiant 1979] In BPP [Jerrum, Sinclair, Vigoda 2004] General graphs: #P-hard

53 Suppose G is a graph with nonnegative edge weights. Planar graphs: Exactly compute PerfMatch(G) In P Fisher, Kasteleyn, Temperley algorithm ε-approximation to PerfMatch(G) In P Bipartite graphs: (permanent of nonnegative matrix) General graphs: #P-hard [Valiant 1979] #P-hard In BPP [Jerrum, Sinclair, Vigoda 2004] [Jerrum Sinclair 1989] Algorithm with runtime poly( V, ε, 1 R) R = NearPerfMatch(G) PerfMatch(G)

54 III. Algorithm

55 Reduction to perfect matchings Theorem There is an (efficiently computable) graph G with positive edge weights, such that and Z β ε PerfMatch(G) NearPerfMatch(G) PerfMatch(G) = O(poly β, n, ε 1 ) We then use [Jerrum, Sinclair 1989] which gives an efficient algorithm for approximating the perfect matching sum.

56 Proof sketch: Start with a Trotter-Suzuki style approximation Tr e βh ε Tr(G J G 2 G 1 ) J = poly(n, β, ε 1 )

57 Proof sketch: Start with a Trotter-Suzuki style approximation Tr e βh ε Tr(G J G 2 G 1 ) J = poly(n, β, ε 1 ) Each G j satisfies G j = e sh+o(s2 ) s > 0 where h is one of the terms in the Hamiltonian

58 Proof sketch: Start with a Trotter-Suzuki style approximation Tr e βh ε Tr(G J G 2 G 1 ) J = poly(n, β, ε 1 ) Each G j satisfies G j = e sh+o(s2 ) s > 0 where h is one of the terms in the Hamiltonian h = Y i Y j X i X j, or h = Y i Y j X i X j, or h = ±(I + Z i )

59 Proof sketch: Start with a Trotter-Suzuki style approximation Tr e βh ε Tr(G J G 2 G 1 ) J = poly(n, β, ε 1 ) Each G j satisfies G j = e sh+o(s2 ) s > 0 where h is one of the terms in the Hamiltonian h = Y i Y j X i X j, or h = Y i Y j X i X j, or h = ±(I + Z i ) The resulting G j are very special gates

60 Proof sketch: Start with a Trotter-Suzuki style approximation Tr e βh ε Tr(G J G 2 G 1 ) J = poly(n, β, ε 1 ) Each G J is from the gate set containing 1-qubit gates t t > 0 and two qubit gates 1 + t t t t > 0 Matchgates

61 Let Γ be a a weighted graph with special input and output edges (k of each, say) Example: t t Input edges Output edges

62 Let Γ be a a weighted graph with special input and output edges (k of each, say) We say Γ implements a k-qubit operator G if y G x = PerfMatch(Γ xy ) Example: Γ xy = remove input edges with x i = 0 and output edges with y i = 0. Require that a perfect matching includes the remaining external edges. t t Input edges Output edges

63 Let Γ be a a weighted graph with special input and output edges (k of each, say) We say Γ implements a k-qubit operator G if y G x = PerfMatch(Γ xy ) Example: Γ xy = remove input edges with x i = 0 and output edges with y i = 0. Require that a perfect matching includes the remaining external edges. x = 00, y = 11 t t t t Input edges Output edges 11 G 00 = t

64 Let Γ be a a weighted graph with special input and output edges (k of each, say) We say Γ implements a k-qubit operator G if y G x = PerfMatch(Γ xy ) Example: Γ xy = remove input edges with x i = 0 and output edges with y i = 0. Require that a perfect matching includes the remaining external edges. t t G = 1 + t t t Input edges Output edges

65 Matchgates compose nicely Γ Implements a 2 qubit gate G Γ Γ Implements G 2

66 Matchgates compose nicely Γ Implements a 2 qubit gate G Γ Implements G 12 G 23 Γ

67 Matchgates compose nicely Γ Implements a 2 qubit gate G Γ Implements Tr(G)

68 Proof sketch: Start with a Trotter-Suzuki style approximation Tr e βh ε Tr(G J G 2 G 1 ) J = poly(n, β, ε 1 ) Each G J is a matchgate.

69 Proof sketch: Start with a Trotter-Suzuki style approximation Tr e βh ε Tr(G J G 2 G 1 ) J = poly(n, β, ε 1 ) Each G J is a matchgate. Matchgate implementable with n input edges and n output edges

70 Proof sketch: Start with a Trotter-Suzuki style approximation Tr e βh ε Tr(G J G 2 G 1 ) J = poly(n, β, ε 1 ) Each G J is a matchgate. Matchgate implementable with n input edges and n output edges Trace is a matchgate with no external edges, i.e., a perfect matching sum.

71 Proof sketch: Start with a Trotter-Suzuki style approximation Tr e βh ε Tr(G J G 2 G 1 ) J = poly(n, β, ε 1 ) Each G J is a matchgate. Matchgate implementable with n input edges and n output edges Trace is a matchgate with no external edges, i.e., a perfect matching sum. (nonplanar and nonbipartite)

72 Proof sketch: Start with a Trotter-Suzuki style approximation Tr e βh ε Tr(G J G 2 G 1 ) J = poly(n, β, ε 1 ) Each G J is a matchgate. This gives first part of theorem: Matchgate implementable with n input edges and n output edges Trace is a matchgate with no external edges, i.e., a perfect matching sum. (nonplanar and nonbipartite) Z β ε PerfMatch(G)

73 Bounding NearPerfMatch(G) We need to show: NearPerfMatch(G) PerfMatch(G) = O(poly β, n, ε 1 ) Recall that a nearly perfect matching is like a perfect matching but with 2 vertices unmatched.

74 Bounding NearPerfMatch(G) We need to show: NearPerfMatch(G) PerfMatch(G) = O(poly β, n, ε 1 ) Recall that a nearly perfect matching is like a perfect matching but with 2 vertices unmatched. NearPerfMatch G = u,v G Ω u,v Ω u,v = sum of nearly perfect matchings with u, v unmatched.

75 Bounding NearPerfMatch(G) We need to show: NearPerfMatch(G) PerfMatch(G) = O(poly β, n, ε 1 ) Recall that a nearly perfect matching is like a perfect matching but with 2 vertices unmatched. NearPerfMatch G = u,v G Ω u,v Ω u,v = sum of nearly perfect matchings with u, v unmatched. To complete the proof we show that Ω u,v PerfMatch(G) Tr G JG J 1 G j OG j 1 G j 2 G i PG i 1 G i 2 G 2 G 1 Tr G J G 2 G 1 = O(1) Imaginary time spin-spin correlation function

76 Open questions Can QMC be used to efficiently simulate quantum adiabatic algorithms with stoquastic Hamiltonians? Other models? See e.g., [Piddock Montanaro 2015]:

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