Analog vs Digital Simulations and the Effects of Trotterization

Jon Pajaud

Two Classes of Computation

• Traditional computations but through quantum means• Factoring

• FFT

• Using quantum systems to simulate quantum systems• Simulating outcomes of N-body systems (more examples) try to get a minute

Outline

• Definitions

• Universal Control

• Examples and Implementations

• Trotter’s Formula

• Quantum Chaos

Analog vs Digital

• Analog has continuous gate

• Digital has a discrete set of gates

• Digital admits fault tolerance because of the discretization but may require infinite steps to approach infinite precision

Analog Quantum Simulators

• Simplest model is emulation where control Hamiltonian is analogous to the target Hamiltonian

• Arbitrary analog simulators do not require analogies but have a set of applied functions that are continuous

• The map between computational basis and system basis is arbitrary

|𝟎𝟎 > |𝟎𝟏 > |𝟏𝟎 > |𝟏𝟏 >

|𝟑 > |𝟏 > |𝟐 > |𝟒 >

Arbitrary Gates

• Continuous control

• 𝐻𝑔𝑎𝑡𝑒 = 𝑓 Ԧ𝜆 , 𝜆𝑖 ∈ ℝ

• The system is controllable if the set of Hamiltonians form a set of generators

• Number of Control parameters > number of free parameters

• Hamiltonian for the Simulator in QuIC B• 𝐻𝑅𝑊 = 𝑓 𝜙𝑥, 𝜙𝑦, 𝜙𝑚𝑤

Physical Setup

Smith, 2013

Hamiltonian in the Lab

Kevin Kuper, SQuInT 2020

Universal Control

Errors in Analog Simulations

• Coherent Errors• Exponential

• Randomized Basis

• State Fidelity

• 𝐹 = < 𝜓𝑇|𝜓 > 2 = 𝑇𝑟 𝜌𝜎 𝜌2

• Incoherent Errors• Weaker

Jessen, 2020

Optimal Control

• How do you go about simulating a system?

• Gradient descent

• Unitary Fidelity

• 𝐹 =1

𝑑𝑖𝑚𝑇𝑟 𝑈†𝑉

• Cost function

• 𝑈 = 𝑓 Ԧ𝜆

• 𝐶 Ԧ𝜆 = 1 −1

𝑑𝑖𝑚𝑇𝑟 𝑓 Ԧ𝜆

†𝑉

Trotter’s Formula

• Baker–Campbell–Hausdorff formula• 𝑒𝐴𝑒𝐵 = 𝑒𝐶

• 𝐶 = 𝐴 + 𝐵 +1

2𝐴, 𝐵 +

1

12𝐴, 𝐴, 𝐵 −

1

12𝐵, 𝐴, 𝐵 + ⋯

• Trotter approximation

• 𝑒−𝑖 𝐻1+𝐻2 𝑡 ≈ lim𝑛→∞

𝑒 Τ−𝑖𝐻1𝑡 𝑛𝑒 Τ−𝑖𝐻2𝑡 𝑛 𝑛

• Suzuki expansions

• 𝑒−𝑖 𝐻1+𝐻2 𝑡 ≈ lim𝑛→∞

𝑒 Τ−𝑖𝐻1𝑡 2𝑛𝑒 Τ−𝑖𝐻2𝑡 𝑛𝑒 Τ−𝑖𝐻1𝑡 2𝑛 𝑛

Simple Example

• 𝐻 = 𝑇 Ƹ𝑝 + 𝑉 Ƹ𝑟

• Discretize position basis

• Potential Term is a position dependent phase

• Can be implemented with ancillary qubits

• Kinetic Term is a momentum dependent phase

• 𝑒−𝑖𝐻𝑡 ≈ 𝑈𝑄𝐹𝑇† 𝑒 Τ−𝑖𝑇𝑡 𝑛 𝑈𝑄𝐹𝑇 𝑒

Τ−𝑖𝑉𝑡 𝑛𝑛

Digital Simulation

• Discrete set of gates

• Sequence of gates as alternative to a waveform

• Trotterization as alternative to optimal control

• New types of behaviors

Lipkin–Meshkov–Glick Model (LMG)

• Hamiltonian• 𝐻 = 𝛼 መ𝐽𝑥 + 𝛽 መ𝐽𝑦

2

• Exact Unitary

• 𝑈𝑡 = 𝑒−𝑖 𝛼 መ𝐽𝑥+𝛽 መ𝐽𝑦2 Τ𝑡 ℏ

• Trotterized Unitary• Quantum Kicked Top (QKT)

• 𝑈𝑡 ≈ 𝑒𝑖𝛼 መ𝐽𝑥 Τ𝑡 𝑛ℏ𝑒𝑖𝛽 መ𝐽𝑦2 Τ𝑡 𝑛ℏ

𝑛

• 𝐻𝑄𝐾𝑇 = 𝛼 መ𝐽𝑥 σ𝑛=0∞ 𝛿(𝑡′ − 𝑛𝜏) + 𝛽 መ𝐽𝑦

2

Classical Phase Space Plots

Nathan Lysne, Winter School 2018

Quantum Phase Space

Nathan Lysne, Winter School 2018

Quantum Phase Space

Chaudhury, 2009

Threshold of Chaos

Sieberer, 2018

A Middle Ground

• Larger step means chaos

• Smaller step means more time for regular control errors to build up

Kevin Kuper, SQuInT 2020

Concluding Remarks

• Analog simulators are not scalable but are useful for studying errors

• Digital simulators are scalable but suffer from Trotter errors

• Trotter errors can introduce chaos

• There is a balance between Trotter errors and implementation errors

References

1. Chaudhury, S., Smith, A., Anderson, B. E., Ghose, S. & Jessen, P. S. Quantum signatures of chaos in a kicked top. Nature 461, 768–771 (2009).2. Deutsch, I. H. Harnessing the Power of the Second Quantum Revolution. (2020).3. Sieberer, L. M. et al. Digital Quantum Simulation, Trotter Errors, and Quantum Chaos of the Kicked Top. 1–20 (2018).4. Suzuki, M. Mathematical Physics Generalized Trotter’s Formula and Systematic Approximants of Exponential Operators and Inner Derivations with Applications to Many-Body Problems. 190, (1976).5. Trotter, H. F. On the Product of Semi-Groups of Operators. Proc. Am. Math. Soc. 10, 545–551 (1959).6. Yung, M. hong, Whitfield, J. D., Boixo, S., Gabriel, D. & Aspuru-guzik, A. Introduction to Quantum Algorithms for Physics and Chemistry. (2016).