A Fault-tolerant Architecture for Quantum Hamiltonian Simulation

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A Fault-tolerant Architecture for Quantum
Hamiltonian Simulation

• Guoming Wang
• Oleg Khainovski

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Background

• Quantum Mechanical Systems are everywhere
• nuclear reaction, chemical molecules,
  superconductor, DNA, ......
• Quantum systems 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

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Hamiltonian

• Shrödingers Equation
• H(t) -- Hamiltonian
• a matrix that represents total energy of the
  system
• usually a sum of local terms

2D Ising Model
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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

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Quantum CAD Flow
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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

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How to Simulate Hamiltonians

• Basic Principle
• Outline

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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
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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

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Optimization by Layering

• Particularly good for Ising Model
• number of layers independent of number of qubits

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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 (gt90 physical operations)
• more gates and movements ? more errors?

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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

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Selective XZ Error Correction

• Observation
• X (Bit flip) and Z (Phase flip) errors have
  different behaviors
• Correct them separately ? further reduce gate
  count

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Datapath Organizations

• Qalypso
• variable sized compute and memory regions,
  ancilla generators,
• teleportation network
• determined based on an analysis of the given
  circuit or users
• choice

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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

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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

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Metric for Probabilistic Computation

• Metric Area-Delay-to-Correct-Result (ADCR)
• For ADCR, lower is better.

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Experimental Results

• Effect of Layering for Ising Model

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Experimental Results

• Effect of Layering for General Hamiltonian

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Experimental Results

• Comparison of Datapaths for General Hamiltonian

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Experimental Results

• Comparison of Datapaths for Random Circuit

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Experimental Results

• Comparison of Error Correction Schemes

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Experimental Results

• Effect of Error Distance Threshold

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Experimental Results

• Overall for General Hamiltonian

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Experimental Results

• Overall for General Hamiltonian

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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?