A Fault-tolerant Architecture for Quantum Hamiltonian Simulation

1. A Fault-tolerant Architecture for Quantum Hamiltonian Simulation Guoming Wang Oleg Khainovski

2. Background • Quantum Mechanical Systems are everywhere • nuclear reaction, chemical molecules, superconductor, DNA, ...... • Quantum system’s states are vectors of exponential length in the number of particles it contains  Large-scale quantum systems are hard to simulate on classical computers • Quantum simulation is the original motivation for building a quantum computer [suggested by Richard Feynman in 1982] • Efficient simulation of quantum systems is perhaps the most important application of quantum computers

3. Hamiltonian • Shrödinger’s Equation • H(t) -- Hamiltonian • a matrix that represents total energy of the system • usually a sum of local terms 2D Ising Model

4. Hamiltonian Simulation • For time-independent Hamiltonian • Time-independent Hamiltonian Simulation Problem • Given the description of a Hamiltonian H and a time t, build a polynomial-size quantum circuit that approximates the unitary transformation

5. Quantum CAD flow Circuit Specification DatapathSynthesis Encoded Circuit Error Correction Synthesis Mapping FU layout & Unmapped Circuit Mapped Circuit Routing Verification Too many failures? Full Layout Success Probability Area & Latency Quantum CAD Flow

6. Quantum CAD flow FU layout & Unmapped Circuit Encoded Circuit Error Correction Synthesis Mapping Circuit Specification Datapath Synthesis Software Hamiltonian Simulation Problem Mapped Circuit Too many failures? Verification Full Layout Routing Success Probability Area & Latency Our Work • Studied the architecture for Hamiltonian Simulation using Quantum CAD flow • A software that, given a Hamiltonian Simulation problem, generates and optimizes the solution circuit, and then feeds it into the CAD flow • Optimizations to the Error Correction Synthesis, Datapath Synthesis and Mapping stages of the CAD flow

7. How to Simulate Hamiltonians • Basic Principle • Outline

8. How to Simulate Hamiltonians • Each local term only acts on few number of qubits, and can be implemented by a relatively small circuit Use Solovay-Kitaev algorithm to find a short sequence of basic instructions

9. Optimization by Layering • Observation: • The ordering of local terms affects the parallelism of the resulting circuit • define “layers” of local terms --- all local terms in the same layer are independent • Term-by-Term  Layer-by-Layer • use a greedy algorithm to find layers

10. Optimization by Layering • Particularly good for Ising Model • number of layers independent of number of qubits

11. Standard Error Correction • Quantum Error Correction • Two Stages: • Correct X (bit flip) error • Correct Z (phase flip) error • Standard Error Correction: place correction after every gate • too expensive (>90% physical operations) • more gates and movements  more errors?

12. Selective Error Correction • Selective Error Correction • place fewer corrections on the Critical Error Path • define an Error Distance Threshold • CNOT propagates the input errors • reduced gate count & satisfactory success probability

13. Selective XZ Error Correction • Observation: • X (Bit flip) and Z (Phase flip) errors have different behaviors • Correct them separately  further reduce gate count

14. Datapath Organizations • Qalypso • variable sized compute and memory regions, ancilla generators, teleportation network • determined based on an analysis of the given circuit or user’s choice

15. Qalypso+ • Idea: reduce the number of expensive long-range teleportation communications • analyze the given circuit, construct a graph whose vertices are data qubits and edges are their interactions • find a relatively small and balanced cut of this graph • each part of data qubits are assigned to a particular compute region as its “favorite” qubits

16. Mapping • For every gate in the program order, decide which functional unit is used to execute it and hence how to move the data qubits • evaluate every functional unit to find the best • if a compute region does not like one of the input qubits, then all the functional units it contains will get a penalty • By this rule, every data qubit lives in a particular compute region most of the time and moves out only when necessary • This partitioning and mapping strategy is particularly good for Hamiltonian Simulation Circuits • particles normally interact only with its neighbors

17. Metric for Probabilistic Computation • Metric: Area-Delay-to-Correct-Result (ADCR) • For ADCR, lower is better.

18. Experimental Results Effect of Layering for Ising Model

19. Effect of Layering for General Hamiltonian Experimental Results

20. Comparison of Datapaths for General Hamiltonian Experimental Results

21. Comparison of Datapaths for Random Circuit Experimental Results

22. Comparison of Error Correction Schemes Experimental Results

23. Effect of Error Distance Threshold Experimental Results

24. Overall for General Hamiltonian Experimental Results

25. Overall for General Hamiltonian Experimental Results

26. Future Direction • Further Improvement to Hamiltonian Simulation • Better Layering? • Better Partitioning and Mapping? • Other aspects: Routing? Ancilla factory? • Extension to Time-dependent Hamiltonian Simulation • Adiabatic algorithms? • Application: studying materials? solving linear systems?