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 Verification Routing Quantum CAD flow Circuit Specification Encoded Circuit FU layout & Unmapped Circuit Mapped Circuit Too many failures?Full Layout Success ProbabilityArea & Latency Mapping Datapath Synthesis Error Correction Synthesis

6. 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 VerificationRouting Quantum CAD flow Circuit Specification Encoded Circuit FU layout & Unmapped Circuit Mapped Circuit Too many failures?Full Layout Success ProbabilityArea & Latency Hamiltonian Simulation Problem SoftwareMapping Datapath Synthesis Error Correction Synthesis

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. Experimental Results  Effect of Layering for General Hamiltonian

20. Experimental Results  Comparison of Datapaths for General Hamiltonian

21. Experimental Results  Comparison of Datapaths for Random Circuit

22. Experimental Results  Comparison of Error Correction Schemes

23. Experimental Results  Effect of Error Distance Threshold

24. Experimental Results  Overall for General Hamiltonian

25. Experimental Results  Overall for General Hamiltonian

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?