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FERMILAB-SLIDES-19-040-QIS This document has been authored by Fermi Research Variational quantum simulation of Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, interacting bosons on NISQ devices Office of High Energy Physics. Andy C. Y. Li Kavli ACP Spring Workshop: Intersections QIS/HEP This manuscript has been authored by Fermi Research Alliance, LLC 20 May 2019 under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of Science, Office of High Energy Physics Andy C. Y. Li Alex Macridin Panagiotis Spentzouris

Outline • Noisy Intermediate-Scale Quantum (NISQ) devices • Quantum-classical hybrid variational algorithms • Variational quantum eigensolver (VQE) of interacting bosons • Proof-of-principle experiment of a 3-qubit implementation • Open questions about scalability 2 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Digital quantum simulation A. A. Houck et al , Nat Phys 8, 292 (2012) control • “Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical” – Richard Feynman (1982) • Digital: all operations are represented by qubit gates • Time evolution, quantum phase estimation, quantum annealing, … • Targets: highly entangled quantum states, Science 334.6052 (2011): 57-61 non-perturbative system dynamics, … 3 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Challenges: noise and control error Pure dephasing ( 𝑈 # ) Relaxation ( 𝑈 " ) • Decoherence: relaxation, pure dephasing, Δ𝐹(𝑢) ←noise correlated noise, … = 𝑏 0 + 𝑐 𝑓 0123 4 4 1 𝜔 𝑢 → device loses ‘quantumness’ after a limited coherence time • Control error: inaccurate gate implementation due to imperfect calibration, qubit drift, … → reliable result only within a limited number of gate operations • Only shallow circuits can be reliably implemented in the near future Martin Savage’s group: PRA 98, 032331 (2018) 4 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Superconducting qubit coherence Phys. Rev. A 76, 042319 (2007) Transmon: simple design with advances in fabrication 2019 and materials 0-pi and other more advanced designs: much more complicated circuit structure Phys. Rev. A 87, 052306 (2013), Phys. Rev. X 3, 011003 (2013) = M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013) 5 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Quantum computing for NISQ devices D-wave quantum annealing https://qutech.nl/ majorana-trilogy- – unclear quantum advantage completed/ Analog open-system simulation Phys. Rev. X 7, Fault-tolerant qubit 011016 (2017) – very limited applications – unclear path to realize Noisy Intermediate- https://ai.googleblog.com/ Scale Quantum 2018/03/a-preview-of- bristlecone-googles- (NISQ) devices: new.html ~100 pre-threshold qubits capable for shallow circuits 6 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Quantum-classical hybrid variational algorithms • Quantum Approximate Optimization Algorithm (QAOA) Cost function – Approximated solutions for combinatorial optimization 𝐷( ⃗ 𝜄) problems through a series of classically optimized gate Classical Hybrid operations computing • Quantum kernel method Evaluate by Evaluate by a Quantum a classical – Support vector machine (SVM) with kernel function device computer evaluated by quantum devices • Quantum autoencoder – encoding in Hilbert space with encoder trained classically Classical optimization • Variational quantum eigensolver (VQE) algorithm – Variational ansatz represented by a list of quantum gate and optimized by a classical optimizer 7 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Why hybrid variational algorithms? • Relatively shallow circuit • Tolerant to control errors (coherent rotation angle errors ) • Quantum advantage? – Heuristic and most likely problem-specific • Possible sources of quantum advantage – Quantum tunneling (QAOA) – Hilbert space size: 2 : (Quantum machine learning) – Natural way to evaluate ⟨𝐼⟩ , … (VQE) 8 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Variational quantum eigensolver (VQE) • Ground-state properties Low-energy Encoding: system • Long-time scale spectrum representation qubits responses Variational ansatz: Noisy intermediate Scale parameterized circuit to Quantum (NISQ) devices prepare the trial state Update Efficient measurement Classical optimization Trial state’s energy algorithm 9 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

VQE applications in quantum chemistry 10 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

VQE: much less developed for non-fermionic systems By en:User_talk:S_kliminUpgrade (New vectorial edition) : Olivier d'ALLIVY KELLY - • Fermions ↔ Qubits en:File:Polaron_scheme1.jpg, CC BY-SA 4.0, https://commons.wikimedia.org/w/i – Jordan-Wigner transformation ndex.php?curid=8461871 • Many models in high-energy and condensed- matter involve non-fermionic degrees of freedom • Goal: many-body systems with bosons • light-matter interaction • electron-phonon coupling 11 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Boson encoding by qubits Goal: encode a truncated boson Hilbert space in qubits Position basis binary encoding Number basis binary encoding Ref: Phys. Rev. Lett. 121, 110504 𝑦 𝑜 = 𝑂 = 1 … 11 @ 𝑦 = Δ :0" = 1 … 11 @ … # Δ 𝑜 = 2 = 0 … 10 @ 𝑦 = Δ :0" = 1 … 10 @ # 0" 𝑜 = 1 = 0 … 01 @ … 𝑜 = 0 = 0 … 00 @ 𝑦 = Δ 0:0" = 0 … 00 @ # 12 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Variational ansatz Motivated by adiabatic state transfer Parameterized gates natively 𝑉 𝛽, 𝜄 = 𝑓 01 H( "0I J K LIJ M ) supported by the hardware Ground Ground state of 𝐼 O state of 𝐼 N |𝜔 ⃗ 𝑉(𝛽 F , 𝜄 F ) 𝑉(𝛽 " , 𝜄 " ) 𝜄 ⟩ |𝜔 ⃗ 𝜄 ⟩ Hardware efficient Model motivated Increasing number of Increasing circuit depth optimization parameters 13 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Cost function for ground state & excited states Ground-state cost function = trial state’s energy 𝐷 F ( ⃗ 𝜄) = 𝜔 ⃗ 𝜄 𝐼 𝜔 ⃗ 𝜄 Ground state: 𝜔 F = argmin 𝐷 F |V(H)⟩ # 1st-excited state cost function: 𝐷 " = 𝜔 ⃗ 𝜄 𝐼 𝜔 ⃗ + 𝜗 𝜔 F |𝜔 ⃗ 𝜄 𝜄 Overlap with the ground state 1st-excited state: 𝜔 " = argmin 𝐷 " |V(H)⟩ # + 𝜗 𝜔 " |𝜔 ⃗ # 2nd-excited state cost function: 𝐷 # = 𝜔 ⃗ 𝜄 𝐼 𝜔 ⃗ + 𝜗 𝜔 F |𝜔 ⃗ 𝜄 𝜄 𝜄 … 14 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Optimization algorithm • Expensive to evaluate gradient of cost function – Numerical differentiation, cost ~ Ο 𝑜 , where 𝑜 = number of parameters – In contrast, for neural network, cost ~ Ο log 𝑜 by back propagation – Preferable: gradient-free optimizer • Noisy cost function – Hardware fidelities, sampling error, … – Preferable: noise insensitive • Local minimums – Low energy but physically very different from the ground state (or targeted state) – Preferable: global optimizer / knowledge to make reasonably good initial guess 15 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Proof-of-principle expt. – Rabi model using Rigetti’ s device TLS Rabi Hamiltonian: two-level system (TLS) coupled to a photon mode Ω 𝐼 = 𝜕𝑏 _ 𝑏 + 𝛻 2 𝜏 b + 𝑕 𝑏 _ + 𝑏 𝜏 c 𝑕 𝜕 Number-basis binary encoding: 𝑜 = 0 = 00 @ 𝑜 = 1 = 01 @ photon mode truncated to up to Photon 𝑜 = 2 = 10 @ 𝑜 = 3 = 11 @ 3 photons 16 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Hardware efficient ansatz Entanglement-gate layers 1Q-gate layer Ansatz consists only of native gates supported by |𝜔 ⃗ 𝜄 ⟩ the hardware e.g. R f (𝜄) , R g (𝜄) and CZ 3 qubits with 1 entanglement layer 17 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics

Optimizers Optimization algorithm Simultaneous Perturbation Stochastic Stochastic Approximation (SPSA) Nelder-Mead Gradient-free Constrained Optimization BY Linear Gradient-free Approximations (COBYLA) Bound Optimization BY Quadratic Gradient-free Approximation (BOBYQA) Covariance Matrix Adaptation Evolutionary algorithm: Evolution Strategy (CMA-ES) stochastic & gradient-free 18 5/20/2019 Andy C. Y. Li | Kavli ACP Spring Workshop: Intersections QIS/HEP at the Aspen Center for Physics