Algorithmic challenges in quantum simulation. Andrew Childs University of Maryland

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1 Algorithmic challenges in quantum simulation Andrew Childs University of Maryland

2 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 Simulating physics with computers MIT Physics of Computation Conference, 1981

3 Computational quantum physics chemical reactions (e.g., nitrogen fixation) condensed matter physics/ properties of materials nuclear/particle physics

4 <latexit sha1_base64="uzqublbdnvctkzw62snnldenram=">aaab/nicbzdnssnafivv6l+tf1hbjzvbirgqiqjqqqi4cvnf2eibymq6aydojmfmipa0c1/fjqsvtz6ho9/gazufth4y+dj3xu6deyscke0431zhyxfpeaw4wlpb39jcsrd37lwcski9evnyngkskgecepppthujpdgkok0h/atxvf5apwkxunodhpor7gowmok1sdr23uxwsswx6hkkltawylltl52kmxgabzehmusqte2vvicmausfjhwr1xsdrpszlportkelvqpogkkfd2ntomarvx42ux+edo3tqweszrmatdzfexmolbpegemmso6p2dry/k/wthv45mdmjkmmgkwxhslhokbjmfchsuo0hxjardjzkyi9ldhrjrksccgd/fi8emev84p7c1ku3uzpfgefduaixdifklxddtwgmirneiu368l6sd6tj2lrwcpndugprm8f63 Implementing quantum algorithms H(0) H(s) A xi = bi H(1) adiabatic optimization exponential speedup by quantum walk evaluating Boolean formulas linear/ differential equations, convex optimization

5 Algorithmic challenges in quantum simulation 1. Efficient simulation with a universal quantum computer 2. Simulating sparse Hamiltonians 3. Simulating quantum mechanics in real time 4. High-precision simulation 5. An optimal tradeoff 6. Real-time simulation revisited 7. Making quantum simulation practical

6 1. Efficient simulation with a universal quantum computer Universal quantum computation The simulation problem The dynamics of a quantum system are 0 U 1 determined by its Hamiltonian H. 0 U 5 U 6 i d dt (t)i = H (t)i U 3 0 U 2 Given a description of the Hamiltonian H, an 0 U 4 evolution time t, and an initial state (0)i, produce the final state (t)i (to within some error tolerance ²). This is as hard as anything a quantum computer can do!

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8 2. Simulating sparse Hamiltonians Sparse Hamiltonians [Aharonov, Ta-Shma 03] For any given row, the locations of the nonzero entries and their values can be computed efficiently H = Main idea: Color the edges of the graph of H. Then the simulation breaks into small pieces that are easy to handle. = + + A sparse graph can be efficiently colored using only local information, so this gives efficient simulations. [Childs, Cleve, Deotto, Farhi, Gutmann, Spielman 03; Aharonov, Ta-Shma 03]

9 3. Simulating quantum mechanics in real time No fast-forwarding theorem: Simulating Hamiltonian dynamics for time t requires (t) gates [Berry, Ahokas, Cleve, Sanders 05] Complexity of kth order product formula simulation is O(5 2k t 1+1/2k ). Can we give an algorithm with complexity precisely O(t)? Systems simulate their own dynamics in real time!

10 <latexit sha1_base64="dboqdlmups5fne6eitraacegnlg=">aaab+nicbvbns8naen3ur1q/yj16wsxcvdrebpvwemsbfywtnkfsttt26wytdydicf0rxjyoepwxeppfug1z0nyha4/3zpizfyaca3ccb6uwtlyyulzcl21sbm3v2lvlex2nijkpxijwrzbojrhkhnaqrjuorqjqsgy4vjz4zuemni/lhywsfkskl3mpuwjg6tjlmyoc+/pbgc8szu linear in t! 3. Simulating quantum mechanics in real time Challenge: mismatch between Schrödinger dynamics (continuous time) and the quantum circuit model (discrete time) Main idea: introduce a discrete-time quantum walk corresponding to the Hamiltonian discrete dynamics easy to implement efficiently carries spectral information about the Hamiltonian Complexity: O(t/ p ) [Childs 10; Berry, Childs 12]

11 <latexit sha1_base64="9w3jkobffuvaorjqbmop8domxfs=">aaaccxicbvbns8naen3ur1q/oh69rbahxtpefpvwemsbfywtngnzblftks0m7g6eenr24l/x4khfq//am//gtzudtj4yelw3w8w8p2zuksv6ngoli0vlk8xv0tr6xuawub1zl6neyolgiewi5snjgoxeuvqx0oofqahpsnmpljo/+ucepbg/u6oyeceacnqngcktdc39g9ena1y57qswbl0ss8oi3kntwjdohhhunctw1zoazhm7j2wqo9e1v9xehjoqciuzkrjtw7hyuiquxyyms24isyxwgaakrsl 4. High-precision simulation? Can we improve the dependence on ²? Many approximate computations can be done with complexity poly(log(1/²)): computing ¼ boosting a bounded-error subroutine Solovay-Kitaev circuit synthesis and more Lower bound (based on the unbounded-error query complexity of parity): log(1/ ) log log(1/ ) Quantum walk simulation: O(1/ p ) Product formulas (kth order): O 5 k / 1/k Can we do better? [Berry, Childs, Cleve, Kothari, Somma 14 & 15]

12 <latexit sha1_base64="pm0yayl/frkk7zirbhp2+8izeri=">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</latexit> sha1_base64="eizgrgsqwa/oi7hufx4aohzc3ve=">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</latexit> sha1_base64="jcij6ulc8fttefrhpsn9galsuma=">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</latexit> 4. High-precision simulation Yes! There is a simulation with complexity O t log(t/ ). log log(t/ ) exponential improvement in ²! Main idea: Consider the truncated Taylor series e iht = P 1 k=0 Expand H as a linear combination of unitary operators ( iht) k k! P K k=0 ( iht) k k! Directly implement the overall linear combination by oblivious amplitude amplification [Berry, Childs, Cleve, Kothari, Somma 14 & 15]

13 5. An optimal tradeoff Combining known lower bounds on the complexity of simulation as a function of t and ² gives t + log 1 log log 1 vs. upper bound of O t log t log log t Recent work, using an alternative method for implementing a linear combination of quantum walk steps, achieves the lower bound. optimal tradeoff between t and ²! Main idea: Encode the eigenvalues of H in a two-dimensional subspace Manipulate those eigenvalues using a carefully-chosen sequence of single-qubit rotations (inspired by quantum control technique of composite pulses) [Low, Chuang 16]

14 6. Real-time simulation revisited We ve focused on the complexity as a function of t (evolution time) and ² (precision). What about the dependence on system size? Consider a system of n spins with nearest-neighbor interactions. To simulate for constant time, best previous methods give: total number of gates: O(n 2 ) circuit depth (execution time with parallel gates): O(n) e ith site index Execution time should not have to be extensive! e ith A Recent improvement: simulation with O(n) gates, O(1) depth e ith Y e ith Y B Main ideas: optimal n dependence! e ith A e ith B Lieb-Robinson bound limits the speed of propagation e ith Y e ith Z Simulate small, overlapping regions, with negative-time evolutions to correct the boundaries e ith Y Z l [Haah, Hastings, Kothari, Low 18]

15 7. Making quantum simulation practical IBM Google/UCSB Delft Maryland When can we use quantum computers to solve problems that are beyond the reach of classical computers? Challenges Improve experimental systems Improve algorithms and their implementation, making the best use of available hardware

16 7. Making quantum simulation practical Factoring a 1024-bit number [Kutin 06] qubits T gates T gates Simulating FeMoco [Reiher et al. 16] 111 qubits T gates Simulating 50 spins (segmented QSP) 67 qubits T gates ,000 10,000 qubits Simulating 50 spins (PF6 empirical) 50 qubits T gates [Childs, Maslov, Nam, Ross, Su 17]

17 Outlook Interplay between physics and computer science Ideas from CS: distributed graph coloring, query complexity lower bounds, Markov chains, Ideas from physics: composite pulses, Lieb-Robinson bounds, scattering, Quantum simulation algorithms will be powerful computational tools for answering questions about quantum physics Ongoing challenges Find faster quantum simulation algorithms that exploit system structure Develop efficient practical implementations Use real quantum computers to do science!

Algorithmic challenges in quantum simulation. Andrew Childs University of Maryland

Algorithmic challenges in quantum simulation Andrew Childs University of Maryland nature isn t classical, dammit, and if you want to make a simulation of nature, you d better make it quantum mechanical,