Einar Pius

Automatic Parallelisation of Quantum Circuits

Using the Measurement Based Quantum

Computing

2Automatic Parallelisation of Quantum Circuits

Motivation

• Quantum computation uses quantum mechanical properties

to represent data and perform operations on it

• Quantum bits (qubits) can be kept stable only for short time

– Due to quantum decoherence

• Algorithms must have as few steps as possible to be able to run on current experimental quantum computers

A two qubit quantum processor created at Yale University in 2009

3Automatic Parallelisation of Quantum Circuits

Goal of this project

• Creating a program that automatically decrease the number

of sequential steps required to perform a quantum

computation

• This was done by applying algebraic transformations to

quantum algorithms

– This may reduce the depth of the quantum computation due to

parallelisation

A two qubit quantum processor created at Yale University in 2009

4Automatic Parallelisation of Quantum Circuits

What we did not do

• Quantum computers do not exist yet

• This was a theoretical project

• No quantum computation was done in this project

A two qubit quantum processor created at Yale University in 2009

5Automatic Parallelisation of Quantum Circuits

Quantum Circuit

• Quantum Circuit is a model of quantum computation

• Qubits are represented by horizontal wires

• Operations on the qubits are represented as gates

6Automatic Parallelisation of Quantum Circuits

Quantum Circuit

• Quantum Circuit is a model of quantum computation

• Qubits are represented by horizontal wires

• Operations on the qubits are represented as gates

• The gates are applied sequentially from left to right

• Gates on the distinct qubits can be applied in parallel

7Automatic Parallelisation of Quantum Circuits

Project Goal

• Transform a quantum circuit to an equivalent quantum circuit

whose depth is less than or equal to the original depth

8Automatic Parallelisation of Quantum Circuits

The parallelisation process

• Translation to the Measurement Based Quantum Computing

(MBQC) model

• Optimisations on MBQC representation

– Standardisation– Signal sifting– Pauli resetting

• Translation back to quantum circuit

• Optimisations on the final circuit

• Result:

– In general the depth of the circuit increases by a log(n) factor– For some circuits the computational depth decreases

9Automatic Parallelisation of Quantum Circuits

A new algorithm

• Translation to MBQC model

• Optimisations on MBQC representation

– Standardisation– Signal sifting– Pauli resetting

• Translation back to quantum circuit

• Optimisations on the final circuit

• We created a new iterative algorithm that translates the

quantum circuits to MBQC model and optimises them

– Runtime O(n³)

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Automatic Parallelisation of Quantum Circuits

The implementation

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Automatic Parallelisation of Quantum Circuits

The implementation

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Automatic Parallelisation of Quantum Circuits

Experiments with the program

• The Toffoli staircase circuit

– Depth will decrease by a constant amount

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Automatic Parallelisation of Quantum Circuits

Experiments with the program

• The Toffoli + CNOT staircase circuit

– The depth of the parallelised circuit will be

constant

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Automatic Parallelisation of Quantum Circuits

Experiments with the program

• A new set of gates

– Every circuit consisting of only the following gates:– The CNOT gate– The ∧Z gate– The ω gate– The phase gate Z(α)– The J(-π/2) gate

– These circuits can be parallelise to a logarithmic depth

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Automatic Parallelisation of Quantum Circuits

Results

• A program for automatic parallelisation of quantum circuits

was created

• A new O(n³) algorithm for translating the quantum circuits to

an optimised MBQC computation was designed

• Three new classes of quantum circuits that could benefit

from the implemented parallelisation method were found

– The Toffoli circuit– The Toffoli + CNOT circuit– A set of gates consisting of CNOT, ∧Z, ω, Z(α), J(-π/2) gates