Automatic Parallelisation of Quantum Circuits Using the Measurement Based Quantum Computing

1. Automatic Parallelisation of Quantum Circuits Using the Measurement Based Quantum Computing Einar Pius

2. 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 Automatic Parallelisation of Quantum Circuits

3. 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 Automatic Parallelisation of Quantum Circuits

4. 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 Automatic Parallelisation of Quantum Circuits

5. Quantum Circuit • Quantum Circuit is a model of quantum computation • Qubits are represented by horizontal wires • Operations on the qubits are represented as gates Automatic Parallelisation of Quantum Circuits

6. 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 Automatic Parallelisation of Quantum Circuits

7. Project Goal • Transform a quantum circuit to an equivalent quantum circuit whose depth is less than or equal to the original depth Automatic Parallelisation of Quantum Circuits

8. 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 Automatic Parallelisation of Quantum Circuits

9. 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) Automatic Parallelisation of Quantum Circuits

10. The implementation Automatic Parallelisation of Quantum Circuits

11. The implementation Automatic Parallelisation of Quantum Circuits

12. Experiments with the program • The Toffoli staircase circuit • Depth will decrease by a constant amount Automatic Parallelisation of Quantum Circuits

13. Experiments with the program • The Toffoli + CNOT staircase circuit • The depth of the parallelised circuit will be constant Automatic Parallelisation of Quantum Circuits

14. 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 Automatic Parallelisation of Quantum Circuits

15. 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 Automatic Parallelisation of Quantum Circuits