Energy lo localiz ization, quantum chaos, , and and th the melt - PowerPoint PPT Presentation

Energy lo localiz ization, quantum chaos, , and and th the melt lt-down of dig igit ital l quantum sim imula lation Philipp Hauke, Heidelberg University Markus Heyl, MPI-PKS Dresden Peter Zoller, IQOQI and University of Innsbruck Trieste, 13.9.2017

Digital quantum simulation could solve important physics problems Condensed matter High-energy (high-T c superconductivity) (QCD...) Temperature Foto: Jonah Bernhard Foto: H Blatt Pressure group Foto: Julian Kelly, Martinis group Foto: Crespi et al., Nat. Photon 2013

Digital quantum simulation approximates time evolution operator by discrete gates Can do Want , Lloyd, Science 1996; Trotter, Proc. Am. Math. Soc. 1959; Suzuki, Prog. Theor. Phys. 1976

Proof-of-principle experiments exist for digital quantum simulation Dynamics of spin models Toy-model lattice gauge theory time Lanyon et al., Science 2011 See also Salathé et al., PRX 2015 Fermionic models particle number time Barends et al., Nat. Comm. 2015 Martinez, Muschik, Schindler, Nigg, Erhard, Heyl, PH, Dalmonte, Monz, Zoller, and Blatt, Nature 2016

How reliable/scalable is that?

Outline Robustness of local observables Worst-case error bound and toy model Numerical results Analytical insights Break-down as transition to quantum chaos Conclusion

Outline Robustness of local observables Worst-case error bound and toy model Numerical results Analytical insights Break-down as transition to quantum chaos Conclusion

Trotterization has a well-controlled error bound Polynomial divergence in t and N (# of qubits) Lloyd, Science 1996 See also Aharonov and Ta-Shma, in Proc. 35th STOC Berry, Ahokas, Cleve, and Sanders, Commun. Math. Phys. 2007 Brown, Munro, and Kendon, Entropy 2010 Childs and Kothari, Lecture Notes in Computer Science 2011

That is a worst case estimate But maybe for our interests that is too much!

Local observables may be much more robust than the total unitary Toy model: trivial time evolution z time y h x

If the field is modified, unitary changes very fast Error in unitary Error in magnetization Independent of N ! Only short times and small systems! N= 32 16 8 4 0.4 0.3 0.2 0.1 0.5 1.0 1.5 time

What about digital quantum simulation and quantum many-body systems?

Outline Robustness of local observables Worst-case error bound and toy model Numerical results Analytical insights Break-down as transition to quantum chaos Conclusion

Outline Robustness of local observables Worst-case error bound and toy model Numerical results Analytical insights Break-down as transition to quantum chaos Conclusion

Numerical example: Ising chain in transverse and longitudinal field h J J J z g y x time Z X Z X Z Z Z Z Z Z X X Z Z Z Z Z Z Z X Z X Z Z Z X Z X

Characterization through energy as an emergent constant of motion † 𝑢 𝐼 𝑉 𝜐 𝑢 |𝜔 0 In Trotterized evolution: 𝐹 𝜐 𝑢 = 𝜔 0 | 𝑉 𝜐 𝑜 𝑢 𝑢 𝑉 𝜐 𝑢 = 𝑉 𝑜 𝑢 = 𝑉 1 𝑜 = 𝜐 . . . 𝑉 𝑁 𝑜 = 𝜐 𝐹 𝜐=0 𝑢 = 𝜔 0 | 𝑓 𝑗 𝐼 𝑢 𝐼𝑓 − 𝑗 𝐼 𝑢 |𝜔 0 = const Ideally: Simulator fidelity: Heating above Ideally: ideal evolution 𝑅 = 0 𝑅(𝑢) = 𝐹 𝜐 (𝑢) − 𝐹 𝜐=0 𝐹 𝑈=∞ − 𝐹 𝜐=0 Normalized to Worst case: 𝑅 = 1 infinite heating

At small Trotter step, local observables become robust 𝑅(𝑢 = ∞) infinite heating 𝑅(𝑢) = 𝐹 𝜐 (𝑢) − 𝐹 𝜐=0 𝐹 𝑈=∞ − 𝐹 𝜐=0 ideal evolution, H conserved quantity Trotter step size perturbative 𝜐 = 𝑢 regime 𝑜 Compare Lloyds bound

Not only the energy, also other local observables become robust magnetization at 𝑢 = ∞ ideal evolution infinite heating Trotter step size

Where does that come from?

Outline Robustness of local observables Worst-case error bound and toy model Numerical results Analytical insights Break-down as transition to quantum chaos Conclusion

Outline Robustness of local observables Worst-case error bound and toy model Numerical results Analytical insights Break-down as transition to quantum chaos Conclusion

Interpret Trotter sequence as periodic driving 2 t / n 3 t / n 4 t / n t / n 5 t / n Period: 𝜐 = 𝑢/𝑜 small expansion parameter 𝜕 = 2𝜌 Frequency: 𝜐 Cold-atom context: e.g., Goldman and Dalibard, PRX 2014 Reviews: Eckardt 2016, Holthaus 2016

Classical analogue: Kaptiza’s pendulum https://youtu.be/rwGAzy0noU0 fast drive slow drive stable unstable Nice comparison classical/quantum: D'Alessio, Polkovnikov, Ann. Phys. 2013

For small period t / n = τ , effective Hamiltonian has a perturbative expansion (Magnus) For small t / n = τ Lloyds bound Cold-atom context: e.g., Goldman and Dalibard, PRX 2014 Reviews: Eckardt 2016, Holthaus 2016

Magnus expansion ensures energy localization D’Alessio and Polkovnikov, Annals of Physics 2013 Zeroth order = time average = target H For small 𝜐 : → emergent constant of motion permits perturbation theory large frequency / small frequency / small Trotter step 𝜐 = 𝑢/𝑜 large Trotter step 𝜐 Q ( t = ∞ ) period # period # τ

Energy localization enables linear response theory 𝐼 1 𝐼 1 𝐼 1 Periodic sequence of two gates 𝐼 1 , 𝐼 2 𝐼 2 𝐼 2 𝐼 2 𝐼 𝑢 = 1 2 𝐼 1 + 𝐼 2 + 1 2 square wave ∗ 𝐼 1 − 𝐼 2 time perturbation at frequency 𝜕 = 2𝜌 𝜐 Main assumption of LRT: ensured by state remains close to unperturbed state energy localization Kubo 1962 ∆𝐶 𝑢 = 𝐶 𝜐 𝑢 − 𝐶 𝜐=0 (𝑢) Consequence: ∆𝐶 ∞ = −𝑗 𝜐 4 Tr(𝜍 0 [𝐶, 𝐼 2 − 𝐼 1 ]) observables deviate only perturbatively magnetization at 𝑢 = ∞ 𝜐

From these analytical arguments, we understand very well the perturbative regime particle number time Martinez et al., Nature 2016 τ If the system is energy localized, this regime is robust Challenge: D’Alessio and Polkovnikov, Ann. Phys. 2013 D'Alessio and Rigol, PRX 2014 Predict breakdown point Lazarides, Das, and Moessner, PRE 2014 Ponte, Papic, Huveneers, and Abanin, PRL 2015 Bukov, Heyl, Huse, and Polkovnikov, PRB 2016 . . .

There may be three different regimes 𝜕 𝜕 ≫ Σ 𝐼 𝑚 Perturbative response No heating at large times many-body 𝜇 𝐼 𝑛 , 𝐼 𝑜 𝜇 ′ spectrum ∀𝜇, 𝜇 ′ , 𝑛, 𝑜 Σ 𝐼 𝑚 > 𝜕 ≫ , 𝐹 𝜇 − 𝐹 𝜇 ′ Perturbative at low order, break down at higher orders possible At most logarithmic heating over time (we do not see this) Abanin, Roeck, and Huveneers, PRL 2015 𝜇 𝐼 𝑛 , 𝐼 𝑜 𝜇 ′ 𝜕 ≫ 𝐹 𝜇 − 𝐹 𝜇 ′ Non-perturbative; infinite heating

Outline Robustness of local observables Worst-case error bound and toy model Numerical results Analytical insights Break-down as transition to quantum chaos Conclusion

Outline Robustness of local observables Worst-case error bound and toy model Numerical results Analytical insights Break-down as transition to quantum chaos Conclusion

Transition to quantum chaos in periodically driven single-particle systems See book Fritz Haake 𝛪 Kicked rotor energy large periods: diffusion, chaos small periods: localization in energy period

Break-down as transition of Floquet Hamiltonian to quantum chaos 𝑉 (1) = = 𝑓 −𝑗 𝐼 𝐺 𝜐 Floquet Hamiltonian NB: eigenvalues follow Wigner-Dyson statistics for all 𝜐 (for generic ideal H ) Characterize chaos through spread over Floquet basis states PR = Σ| 𝜔 0 |𝜒 𝜉 | 4 𝜇 PR = −log(PR)/N ۧ | 𝜒 𝜉 = eigenstates of Floquet Hamiltonian all basis states are equally likely

Outline Robustness of local observables Worst-case error bound and toy model Numerical results Analytical insights Break-down as transition to quantum chaos Conclusion

Outline Robustness of local observables Worst-case error bound and toy model Numerical results Analytical insights Break-down as transition to quantum chaos Conclusion

Conclusion Digital quantum simulators are more robust than one may think (for local observables) Sharp threshold, connected to quantum chaos We understand the perturbative behavior from periodically driven systems and LRT Valid also for Trotter on classical computers (e.g. tensor networks) Paper in preparation!