Hamiltonian simulation with nearly optimal dependence on all parameters

Transcription

1 Hamiltonian simulation with nearly optimal dependence on all parameters Dominic Berry + Andrew Childs obin Kothari ichard Cleve olando Somma

2 Quantum simulation by quantum walks Dominic Berry + Andrew Childs obin Kothari ichard Cleve olando Somma

3 Simulation of Hamiltonians

4 Outline History of quantum simulation Definition of problem Main result Standard method New techniques Quantum walks (2012) Compressed product formulae (2013) Implementing Taylor series (201) Superposition of quantum walk steps (201)

5 Simulation of Hamiltonians ichard Feynman 1981: Idea of quantum computer Seth Lloyd 1996: Algorithm to simulate interaction Hamiltonians Aharonov + Ta-Shma 2003: Algorithm to simulate sparse Hamiltonians

6 Simulation of Hamiltonians Aharonov + Ta-Shma 2003: Algorithm to simulate sparse Hamiltonians Harrow, Hassidim, Lloyd 2009: Quantum algorithm to solve linear systems Childs, Cleve, Jordan, Yonge-Mallo 2009: Quantum algorithm for NAND trees Berry 201: Quantum algorithm for differential equations Clader, Jacobs, Sprouse 2013: Quantum algorithm for scattering problems

7 The simulation problem Problem: Given a Hamiltonian H, simulate d ψ = ih(t ) ψ dt for time t and error no more than ε. Inputs: H, t and ε. Parameters of H: Output: d Final sparseness state ψ encoded on qubits. N dimension H norm of the Hamiltonian H norm of the time-derivative

8 Main result O τ polylog τ = d H max t Queries: O τ log(τ/ε) log log(τ/ε) Gates: O τ log2 (τ/ε) log log(τ/ε)

9 Comparison to prior work O τ polylog τ = d H max t 1. Lloyd 1996: poly d, log N Ht 2 /ε 2. Aharonov & TaShma 2003: poly d, log N Ht 3/2 /ε 1/2 3. Berry, Cleve, Ahokas, Sanders 2007: d Ht log N 1+δ 1/ε δ. Childs & Kothari 2011: d 3 Ht log N 1+δ 1/ε δ 5. Berry & Childs 2012: d H max t/ε 1/2 6. Berry, Childs, Cleve, Kothari, Somma 2013: d 2 H max t polylog

10 Comparison to lower bound Upper bound: O τ polylog τ = d H max t Lower bound: O τ + polylog τ = d H max t

11 Model Local interactions H = H 1 + H 2 + H 3 + H +

12 Model Sparse Hamiltonians Query: An efficient algorithm to determine the positions and values of non-zero entries. Includes local interactions as a special case.

13 Standard method Use decomposition as H = M k=1 H k Approximate evolution for short time as e iht M e ih kt k=1 For longer times, divide up into many short times e iht r M e ih kt/r k=1 S. Lloyd, Science 273, 1073 (1996).

14 Advanced methods A. Quantum walks (2012) B. Compressed product formulae (2013) / Implementing Taylor series (201) C. Superposition of quantum walk steps (201) D. W. Berry and A. M. Childs, Quantum Information and Computation 12, 29 (2012). D. W. Berry, A. M. Childs,. Cleve,. Kothari,. D. Somma, arxiv: (2013).

15 Quantum walks Classical walk: position x jumps either to the left or the right at each step. Quantum walk has position and coin values x, c It then alternates coin and step operators, C x, ±1 = x, 1 ± x, 1 / 2 S x, c = x + c, c The position can progress linearly in the number of steps.

16 Quantum walks Classical walk: position x jumps either to the left or the right at each step. Quantum walk has position and coin values x, c It then alternates coin and step operators, C x, ±1 = x, 1 ± x, 1 / 2 S x, c = x + c, c The position can progress linearly in the number of steps. Szegedy quantum walk allows arbitrary dimensions, n and m on the two subsystems. Szegedy quantum walk uses more general controlled diffusion operators.

17 Szegedy quantum walk The diffusion operators are of the form 2CC I 2 I C is controlled by the first register and acts on the second register. The operator C is a controlled reflection. n C = i i c i i=1 m c i = c[i, j] j j=1 c i The diffusion operator 2 I is controlled by the second register and acts on the first. M. Szegedy, quant-ph/ (200).

18 A. M. Childs, Commun. Math. Phys. 29, 581 (2009). Szegedy walk for Hamiltonians Use symmetric system, with n = m and c i, j = r i, j = H ij The step of the quantum walk is (S is swap) V = is(2cc I) Eigenvalues and eigenvectors are related to those of Hamiltonian. We need to modify to lazy quantum walk, with c i = δ N H 1 j=1 H ij j + 1 σ iδ H 1 N + 1 σ i N j=1 H ij extra component

19 State preparation Grover state preparation starts from 1 N N k=1 k otate ancilla according to amplitude for state to be prepared ψ b N = 1 N k=1 k ψ k ψ k 2 1 Amplitude amplification yields component where ancilla is zero. In comparison, state we wish to prepare is c i = δ N H 1 j=1 H ij j + 1 σ iδ H 1 N + 1 We can just use one iteration! D. W. Berry and A. M. Childs, Quantum Information and Computation 12, 29 (2012). L. K. Grover, PL 85, 133 (2000).

20 Szegedy walk for Hamiltonians Three step process: 1. Start with state in one of the subsystems, and perform controlled state preparation. 2. Perform steps of quantum walk to approximate Hamiltonian evolution. V 3. Invert controlled state preparation, so final state is in one of the subsystems. A. M. Childs, Commun. Math. Phys. 29, 581 (2009).

21 A. M. Childs, Commun. Math. Phys. 29, 581 (2009). Szegedy walk for Hamiltonians A Hamiltonian H has eigenvalues λ. V is the step of a quantum walk, and has eigenvalues ±i arcsin λδ μ ± = ±e π arcsin λδ arcsin λδ λδ We aim to achieve evolution under the Hamiltonian. It has eigenvalues e iλt μ μ +

22 Advanced methods A. Quantum walks (2012) B. Compressed product formulae (2013) / Implementing Taylor series (201) C. Superposition of quantum walk steps (201) D. W. Berry and A. M. Childs, Quantum Information and Computation 12, 29 (2012). D. W. Berry, A. M. Childs,. Cleve,. Kothari,. D. Somma, arxiv: (2013).

23 Compressed product formulae 1. Decompose Hamiltonian into a sum of self-inverse Hamiltonians. 2. Approximate Hamiltonian evolution by Lie-Trotter formula, then compress it. 3. Use oblivious amplitude amplification. m qubits ψ U 2 U 1 U 1 U 2 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 measure b 1 b 2 b 3 b... b m ψ t

24 Compressed product formulae 1. Decompose Hamiltonian into a sum of self-inverse Hamiltonians. 2. Approximate Hamiltonian 1. Decompose evolution Hamiltonian by Lie-Trotter into 1-sparse. formula, then compress 2. Break it. 1-sparse into X, Y, Z parts. 3. Break b X, Y, Z parts into self-inverse. 1 b 2 b 3. Use oblivious 3 b amplitude m qubits.. amplification.. ψ U 2 U 1 U 1 U 2 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 measure b m ψ t

25 Decompose Hamiltonian to 1-sparse Decompose Hamiltonian into H 1 and H 2 : No more than d nonzero elements in any row or column. In general can decompose into d 2 parts.

26 Decompose Hamiltonian to 1-sparse Decompose Hamiltonian into H 1 and H 2 : 0 H No more than d nonzero elements in any row or column. In general can decompose into d 2 parts.

27 Decompose 1-sparse to X, Y, Z Break into X, Y and Z components: H off-diagonal real off-diagonal imaginary on-diagonal real 0 + break into γ-size pieces to get self-inverse

28 Net result M H = γ U j j=1

29 Evolution using control qubits H 1 = γu 1 U 1 is self-inverse 0 P 0 ψ U 1 e ih 1δt ψ = α β β α P = i. Cleve, D. Gottesman, M. Mosca,. Somma, and D. Yonge-Mallo, In Proc. 1st ACM Symposium on Theory of Computing, pp (2009).

30 Simulation of segments m qubits measure b 1 b 2 b 3 b... b m ψ U 2 U 1 U 1 U 2 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 ψ t

31 Simulation of segments m qubits measure b 1 b 2 b 3 b... b m ψ U 2 U 1 U 1 U 2 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 U 2 U 1 ψ p 1 p 2 p 3

32 Simulation of segments m qubits ψ U j U j U j U j U j p 1 p 2 U j U j U j U j U j U j U j U j U j U j U j measure b 1 b 2 b 3 b... b m ψ Mostly the identity, so can be omitted. p 3 U j {U 1, U 2, I}

33 Simulation of segments m qubits measure b 1 b 2 b 3 b... b m ψ U j U j U j ψ p 1 p 2 p 3 U j {U 1, U 2, I}

34 Simulation of segments m qubits measure b 1 b 2 b 3 b... b m ψ U j U j U j ψ p 1 p 2 p 3 U j {U 1, U 2, I}

35 Compression of control qubits m qubits measure b 1 b 2 b 3 b... b m ψ U j U j U j ψ p 1 p 2 p 3 Compressed form: c x = p 1 p x m k x

36 Compression of control qubits c 0 m W? measure 0 m success ψ U j U j U j ψ p 1 p 2 p 3 Compressed form: c x = p 1 p x m k x

37 Oblivious amplitude amplification c 0 m W W ψ U j U j U j ψ t

38 Oblivious amplitude amplification W W success! ψ U j U j U j ψ t

39 Oblivious amplitude amplification W W ψ U j U j U j ψ U 0 ψ = p 0 V ψ + 1 p 1 φ Operation we know how to perform Operation we want to perform Standard amplitude amplification: Need to reflect about U 0 ψ.

40 Oblivious amplitude amplification W W ψ U j U j U j ψ U 0 ψ = p 0 V ψ + 1 p 1 φ Operation we know how to perform Operation we want to perform Standard amplitude amplification: Need to reflect about U 0 ψ.

41 Oblivious amplitude amplification W W ψ U j U j U j ψ U 0 ψ = p 0 V ψ + 1 p 1 φ Operation we know how to perform Operation we want to perform Oblivious amplitude amplification: Only do reflections on first register.

42 Advanced methods A. Quantum walks (2012) B. Compressed product formulae (2013) / Implementing Taylor series (201) C. Superposition of quantum walk steps (201) D. W. Berry and A. M. Childs, Quantum Information and Computation 12, 29 (2012). D. W. Berry, A. M. Childs,. Cleve,. Kothari,. D. Somma, arxiv: (2013).

43 Implementing Taylor series The Hamiltonian evolution can be expanded in Taylor series: U = exp iht = 1 k=0 k! iht k For r segments, we would want U r = exp iht/r K 1 k=0 k! iht/r k

44 Implementing Taylor series If H is unitary, can probabilistically implement using controlled operation. K 1 k=0 k! it/r k k 0 P 0 ψ H k e iht/r ψ

45 Implementing Taylor series In reality H is (approximately) a sum of unitaries H γ M U l l=1 Exponential is then exp iht/r K M k l 1 =1 M l 2 =1 M l k =1 it/r k k! U l1 U l2 U lk We can again implement using controlled operations.

46 Implementing a Taylor series k l 1 W W l 2 l K measure ψ U l1 U l2 U lk ψ A measurement result of 0 corresponds to success. This can be performed deterministically using oblivious amplitude amplification.

47 Advanced methods A. Quantum walks (2012) B. Compressed product formulae (2013) / Implementing Taylor series (201) C. Superposition of quantum walk steps (201) D. W. Berry and A. M. Childs, Quantum Information and Computation 12, 29 (2012). D. W. Berry, A. M. Childs,. Cleve,. Kothari,. D. Somma, arxiv: (2013).

48 Superposition of quantum walk A Hamiltonian H has eigenvalues λ. V is the step of a quantum walk, and has eigenvalues ±i arcsin λδ μ ± = ±e π arcsin λδ arcsin λδ λδ We aim to achieve evolution under the Hamiltonian. It has eigenvalues e iλt μ μ +

49 Superposition of quantum walk A Hamiltonian H has eigenvalues λ. V is the step of a quantum walk, and has eigenvalues ±i arcsin λδ μ ± = ±e π arcsin λδ arcsin λδ λδ We aim to achieve evolution under the Hamiltonian. It has eigenvalues e iλt μ μ + Corrected step V c has eigenvalues μ = e i arcsin λδ

50 Superposition of quantum walk We have We aim for μ = e i arcsin λδ e iλt Try superposition of operations V sup = K k=0 α k V c k 0 W W 0 ψ V c k V sup ψ

51 Solving for α k We have We aim for μ = e i arcsin λδ e iλt Try superposition of operations V sup = K k=0 α k V c k The eigenvalues of V sup are μ sup = K k=0 α k μ k We can solve for α k such that μ sup = e itλ + O( tλ K+1 ) Symmetry is better: μ sup = K k= K α k μ k Then we can get μ sup = e itλ + O( tλ 2K+1 )

52 Analytic formula for α k The generating function for Bessel functions is: k= J k (x)μ k = exp x 2 μ 1 μ For μ = e i arcsin λδ this gives us what we want: exp x 2 μ 1 μ = e ixλδ J k (1) Take x = t/δ to get e iλt. k

53 Analytic formula for α k The generating function for Bessel functions is: k= J k (x)μ k = exp x 2 μ 1 μ For μ ± = ±e ±i arcsin λδ we have: exp x 2 μ ± 1 = e ixλδ μ ± J k (1) Take x = t/δ to get e iλt. k

54 The complete algorithm Map into doubled Hilbert space. Divide the time into d H max t segments. 0 For each segment: 1. Perform the superposition. 0 W W 0 ψ J k (x) V k V sup ψ t 2. Use amplitude amplification to obtain success deterministically. Map back to original Hilbert space. Total complexity: d H max t K

55 Choosing the value of K Bessel function may be bounded as J k x 1 x k k! 2 Scaling is the same as for Taylor series! We can choose K to be polylog K log(τ/ε) log log(τ/ε) J k (1) Overall scaling is O(d H max t polylog) k

56 Single-segment approach J k (d H max t) polylog k d H max t

57 Conclusions We have complexity of sparse Hamiltonian simulation scaling as O(d H max t polylog) The lower bound is scaling as O(d H max t + polylog) The method combines the quantum walk and compressed product formula approaches.