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Ross Duncan ICALP Bordeaux 2010

Rewriting measurement-based

quantum computations with generalised flow

Ross Duncan

Oxford University

Computing Laboratory

Simon Perdrix

CNRS, LIG,

Université de Grenoble

Ross Duncan ICALP Bordeaux 2010

What is the point?

Measurement-based quantum computation

compute by measuring a large entangled state can it be done deterministically? is the given program correct?

Graphical Rewriting

translate MBQC into diagrammatic syntax equational theory based on high-level properties of QM rewrite graphs to translate back to quantum circuits

Ross Duncan ICALP Bordeaux 2010

Z= 10 0-1 X= 01 10

Quantum Measurements

α|0?+β|1?

|0? p=α 2 |1? p=β 2 p=(α+β)/2 2 p=(α-β)/2 2

We can measure the state of qubit

|ψ?=α|0?+β|1?

Ross Duncan ICALP Bordeaux 2010

Quantum Measurements

α|0?+β|1?

|0? p=α 2 |1? p=β 2 p=(α+β)/2 2 p=(α-β)/2 2

We can measure the state of qubit

|ψ?=α|0?+β|1?

Z={|0?,|1?}X={

|0?+|1? 2 |0?+|1? 2

Ross Duncan ICALP Bordeaux 2010

The One-Way Model

Some qubits, each initialised in the state

1 2 1 1

Ross Duncan ICALP Bordeaux 2010

Entangling operation on pairs of qubits to make a graph state. CZ= 1000
0100
0010 000-1

The One-Way Model

Some qubits, each initialised in the state

1 2 1 1

Ross Duncan ICALP Bordeaux 2010

The One-Way Model

A graph state, coupled to some input qubits:The only operation is to measure single qubits in the basis:

Ross Duncan ICALP Bordeaux 2010

The One-Way Model

A graph state, coupled to some input qubits:

* Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state.

Ross Duncan ICALP Bordeaux 2010

The One-Way Model

A graph state, coupled to some input qubits:

* Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state.

Ross Duncan ICALP Bordeaux 2010

The One-Way Model

A graph state, coupled to some input qubits:

* Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state.

Ross Duncan ICALP Bordeaux 2010

The One-Way Model

A graph state, coupled to some input qubits:

* Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state.

Ross Duncan ICALP Bordeaux 2010

The One-Way Model

A graph state, coupled to some input qubits:

* Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state. Finally, any output qubits, they can be corrected by a Pauli Z

Ross Duncan ICALP Bordeaux 2010

The One-Way Model

A graph state, coupled to some input qubits:

* Measured qubits are removed from the cluster; * The outcome of measurement alters the remaining state. Finally, any output qubits, they can be corrected by a Pauli

Ross Duncan ICALP Bordeaux 2010

Non-determinism

Non-determinism of measurements leads to probabilistic branching

Yay!Boo!

Yay!Boo!

Yay!Boo!Yay!Boo!Yay!Boo!Yay!Boo!

Attempt to control branching by using adaptive measurements: (choice of measurements depend on the outcome of earlier ones)

Ross Duncan ICALP Bordeaux 2010

Non-determinism

Non-determinism of measurements leads to probabilistic branching

Yay!Boo!

Yay!Boo!

Yay!Boo!Yay!Boo!Yay!Boo!Yay!Boo!

Attempt to control branching by using adaptive measurements: (choice of measurements depend on the outcome of earlier ones)

Can we carry out measurement-based

computation deterministically?

Ross Duncan ICALP Bordeaux 2010

Determinism in MBQC

Measurement

Pattern

"low-level program"

Ross Duncan ICALP Bordeaux 2010

Determinism in MBQC

Measurement

Pattern

"low-level program"

Geometry

"entangled resource" "graph state" implicitly defines

Ross Duncan ICALP Bordeaux 2010

Determinism in MBQC

Measurement

Pattern

"low-level program"

Geometry

"entangled resource" "graph state" implicitly defines

Flow and GFlow

"correction strategy" can possess

Ross Duncan ICALP Bordeaux 2010

Uniformly

Deterministic

Pattern

jointly determine

Determinism in MBQC

Measurement

Pattern

"low-level program"

Geometry

"entangled resource" "graph state" implicitly defines

Flow and GFlow

"correction strategy" can possess

Ross Duncan ICALP Bordeaux 2010

Uniformly

Deterministic

Pattern

jointly determine

Determinism in MBQC

Measurement

Pattern

"low-level program"

Geometry

"entangled resource" "graph state" implicitly defines

Flow and GFlow

"correction strategy" can possessnot necessarily the same!

Ross Duncan ICALP Bordeaux 2010

Uniformly

Deterministic

Pattern

jointly determine

Determinism in MBQC

Measurement

Pattern

"low-level program"

Geometry

"entangled resource" "graph state" implicitly defines

Flow and GFlow

"correction strategy" can possessnot necessarily the same!

Quantum Circuit

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