Toward the first quantum simulation with quantum speedup Andrew Childs University of Maryland Dmitri Maslov Yuan Su Yunseong Nam Neil Julien Ross arXiv:1711.10980; PNAS 2018
Toward practical quantum speedup IBM Google/UCSB Delft Maryland
Toward practical quantum speedup IBM Google/UCSB Delft Maryland Important early goal: demonstrate quantum computational advantage
Toward practical quantum speedup IBM Google/UCSB Delft Maryland Important early goal: demonstrate quantum computational advantage … but can we find a practical application of near-term devices?
Toward practical quantum speedup IBM Google/UCSB Delft Maryland Important early goal: demonstrate quantum computational advantage … but can we find a practical application of near-term devices? Challenges • Improve experimental systems • Improve algorithms and their implementation, making the best use of available hardware
Toward practical quantum speedup IBM Google/UCSB Delft Maryland Important early goal: demonstrate quantum computational advantage … but can we find a practical application of near-term devices? Challenges • Improve experimental systems • Improve algorithms and their implementation, making the best use of available hardware Our goal: Produce concrete resource estimates for the simplest possible practical application of quantum computers
Quantum simulation “… nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.” Richard Feynman (1981) Simulating physics with computers
Quantum simulation “… nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.” Richard Feynman (1981) Simulating physics with computers Quantum simulation problem: Given a description of the Hamiltonian H , an evolution time t , and an initial state , | ψ (0) i produce the final state (to within | ψ ( t ) i some error tolerance ² )
Quantum simulation “… nature isn’t classical, dammit, A classical computer cannot even represent and if you want to make a the state efficiently. simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.” Richard Feynman (1981) Simulating physics with computers Quantum simulation problem: Given a description of the Hamiltonian H , an evolution time t , and an initial state , | ψ (0) i produce the final state (to within | ψ ( t ) i some error tolerance ² )
Quantum simulation “… nature isn’t classical, dammit, A classical computer cannot even represent and if you want to make a the state efficiently. simulation of nature, you’d better make it quantum mechanical, and A quantum computer cannot produce a by golly it’s a wonderful problem, because it doesn’t look so easy.” complete description of the state. Richard Feynman (1981) Simulating physics with computers Quantum simulation problem: Given a description of the Hamiltonian H , an evolution time t , and an initial state , | ψ (0) i produce the final state (to within | ψ ( t ) i some error tolerance ² )
Quantum simulation “… nature isn’t classical, dammit, A classical computer cannot even represent and if you want to make a the state efficiently. simulation of nature, you’d better make it quantum mechanical, and A quantum computer cannot produce a by golly it’s a wonderful problem, because it doesn’t look so easy.” complete description of the state. Richard Feynman (1981) But given succinct descriptions of Simulating physics with computers • the initial state (suitable for a quantum computer to prepare it efficiently) and Quantum simulation problem: Given a • a final measurement (say, measurements description of the Hamiltonian H , an of the individual qubits in some basis), evolution time t , and an initial state , a quantum computer can efficiently answer | ψ (0) i produce the final state (to within questions that (apparently) a classical one | ψ ( t ) i some error tolerance ² ) cannot.
Product formula simulation L X Suppose we want to simulate H = H ` ` =1 [Lloyd 96]
Product formula simulation L X Suppose we want to simulate H = H ` ` =1 Combine individual simulations with the Lie product formula. E.g., with two terms: e − iAt/r e − iBt/r � r = e − i ( A + B ) t � lim r →∞ [Lloyd 96]
Product formula simulation L X Suppose we want to simulate H = H ` ` =1 Combine individual simulations with the Lie product formula. E.g., with two terms: e − iAt/r e − iBt/r � r = e − i ( A + B ) t � lim r →∞ e − iAt/r e − iBt/r � r = e − i ( A + B ) t + O ( t 2 /r ) � [Lloyd 96]
Product formula simulation L X Suppose we want to simulate H = H ` ` =1 Combine individual simulations with the Lie product formula. E.g., with two terms: e − iAt/r e − iBt/r � r = e − i ( A + B ) t � lim r →∞ e − iAt/r e − iBt/r � r = e − i ( A + B ) t + O ( t 2 /r ) � To ensure error at most ² , take ( k H k t ) 2 / ✏ � � r = O [Lloyd 96]
Product formula simulation L To get a better approximation, use higher-order X Suppose we want to simulate H = H ` formulas. ` =1 E.g., second order: ( e − iAt/ 2 r e − iBt e − iAt/ 2 r ) r = e − i ( A + B ) t Combine individual simulations with the Lie product formula. E.g., with two terms: + O ( t 3 /r 2 ) e − iAt/r e − iBt/r � r = e − i ( A + B ) t � lim r →∞ e − iAt/r e − iBt/r � r = e − i ( A + B ) t + O ( t 2 /r ) � To ensure error at most ² , take ( k H k t ) 2 / ✏ � � r = O [Lloyd 96] [Berry, Ahokas, Cleve, Sanders 07]
Product formula simulation L To get a better approximation, use higher-order X Suppose we want to simulate H = H ` formulas. ` =1 E.g., second order: ( e − iAt/ 2 r e − iBt e − iAt/ 2 r ) r = e − i ( A + B ) t Combine individual simulations with the Lie product formula. E.g., with two terms: + O ( t 3 /r 2 ) e − iAt/r e − iBt/r � r = e − i ( A + B ) t � Systematic expansions to arbitrary order are lim r →∞ known [Suzuki 92] e − iAt/r e − iBt/r � r = e − i ( A + B ) t + O ( t 2 /r ) � Using the 2 k th order expansion, the number of exponentials required for an approximation To ensure error at most ² , take with error at most ² is at most ( k H k t ) 2 / ✏ � � r = O ⌘ 1 / 2 k ⇣ L k H k t 5 2 k L 2 k H k t ✏ [Lloyd 96] [Berry, Ahokas, Cleve, Sanders 07]
Quantum walk simulation [Childs10; Berry, Childs 12]
Quantum walk simulation Quantum walk corresponding to H Alternately reflect about , span {| ψ j i } N j =1 N ✓ ◆ q X , | ψ j i := | j i ⌦ jk | k i + ν j | N + 1 i ν H ∗ k =1 and swap the two registers. [Childs10; Berry, Childs 12]
Quantum walk simulation Quantum walk corresponding to H Alternately reflect about , span {| ψ j i } N j =1 N ✓ ◆ q X , | ψ j i := | j i ⌦ jk | k i + ν j | N + 1 i ν H ∗ k =1 and swap the two registers. If H is sparse, this walk is easy to implement. [Childs10; Berry, Childs 12]
Quantum walk simulation Quantum walk corresponding to H Alternately reflect about , span {| ψ j i } N j =1 N ✓ ◆ q X , | ψ j i := | j i ⌦ jk | k i + ν j | N + 1 i ν H ∗ k =1 and swap the two registers. If H is sparse, this walk is easy to implement. Spectral theorem: Each eigenvalue ¸ of H corresponds to two eigenvalues ± e ± i arcsin λ of the walk operator (with eigenvectors closely related to those of H ). [Childs10; Berry, Childs 12]
Quantum walk simulation Quantum walk corresponding to H Simulation by phase estimation | λ i 7! | λ i | ^ Alternately reflect about , span {| ψ j i } N (phase estimation) arcsin λ i j =1 N 7! e − i λ t | λ i | ^ ✓ ◆ q arcsin λ i X , | ψ j i := | j i ⌦ jk | k i + ν j | N + 1 i ν H ∗ 7! e − i λ t | λ i (inverse phase est) k =1 and swap the two registers. If H is sparse, this walk is easy to implement. Spectral theorem: Each eigenvalue ¸ of H corresponds to two eigenvalues ± e ± i arcsin λ of the walk operator (with eigenvectors closely related to those of H ). [Childs10; Berry, Childs 12]
Quantum walk simulation Quantum walk corresponding to H Simulation by phase estimation | λ i 7! | λ i | ^ Alternately reflect about , span {| ψ j i } N (phase estimation) arcsin λ i j =1 N 7! e − i λ t | λ i | ^ ✓ ◆ q arcsin λ i X , | ψ j i := | j i ⌦ jk | k i + ν j | N + 1 i ν H ∗ 7! e − i λ t | λ i (inverse phase est) k =1 and swap the two registers. If H is sparse, this walk is easy to implement. Theorem: steps of this walk suffice O ( t/ √ ✏ ) to simulate H for time t with error at most ² . Spectral theorem: Each eigenvalue ¸ of H corresponds to two eigenvalues ± e ± i arcsin λ of the walk operator (with eigenvectors closely related to those of H ). [Childs10; Berry, Childs 12]
Taylor series simulation Main idea: Directly implement the series ∞ ( − iHt ) k e − iHt = X k ! k =0 [Berry, Childs, Cleve, Kothari, Somma 14 & 15]
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