Lecture 3: Quantum simulation algorithms

Dominic BerryMacquarie University

We want to simulate the evolution

The Hamiltonian is a sum of terms:

Simulation of Hamiltonians

Seth Lloyd

1996

We can perform

For short times we can use

For long times

For short times we can use

This approximation is because

If we divide long time into intervals, then

Typically, we want to simulate a system with some maximum allowable error .

Then we need .

Simulation of Hamiltonians

Seth Lloyd

1996

Higher-order simulation A higher-order decomposition is

If we divide long time into intervals, then

Then we need . General product formula can give error for time . For time the error is To bound the error as the value of scales as

The complexity is .

2007Berry, Ahokas,

Cleve, Sanders

Higher-order simulation

The complexity is . For Sukuki product formulae, we have an additional factor in

The complexity then needs to be multiplied by a further factor of . The overall complexity scales as

We can also take an optimal value of , which gives scaling

2007Berry, Ahokas,

Cleve, Sanders

Solving linear systems Consider a large system of linear equations:

First assume that the matrix is Hermitian. It is possible to simulate Hamiltonian evolution

under for time : . Encode the initial state in the form

2009

Harrow, Hassidim & Lloyd

The state can also be written in terms of the eigenvectors of as

We can obtain the solution if we can divide each by . Use the phase estimation technique to place the estimate of in an

ancillary register to obtain

Solving linear systems Use the phase estimation technique to place the

estimate of in an ancillary register to obtain

Append an ancilla and rotate it according to the value of to obtain

2009

Harrow, Hassidim & Lloyd

Invert the phase estimation technique to remove the estimate of from the ancillary register, giving

Use amplitude amplification to amplify the component on the ancilla, giving a state proportional to

Solving linear systems

What about non-Hermitian ?

Construct a blockwise matrix

The inverse of is then

This means that

In terms of the state

2009

Harrow, Hassidim & Lloyd

Solving linear systemsComplexity Analysis

We need to examine:1. The complexity of simulating the Hamiltonian to

estimate the phase.2. The accuracy needed for the phase estimate.3. The possibility of being greater than .

2009

Harrow, Hassidim & Lloyd

The complexity of simulating the Hamiltonian for time is approximately .

To obtain accuracy in the estimate of , the Hamiltonian needs to be simulated for time .

We actually need to multiply the state coefficients by , to give

To obtain accuracy in , we need accuracy in the estimate of .

Final complexity is

Differential equations Discretise the differential equation, then encode as a linear system. Simplest discretisation: Euler method.

1j jjA

h

x x

x b

0 in

1

2

3

4

0 0 0 0( ) 0 0 00 ( ) 0 00 0 0 00 0 0 0

II Ah I h

I Ah I hI I

I I

x xx bx bxx

sets initial condition

sets x to be constant

d Adt

x x b

2010Berry

Quantum walks A classical walk has a position which is an

integer, , which jumps either to the left or the right at each step.

The resulting distribution is a binomial distribution, or a normal distribution as the limit.

The quantum walk has position and coin values

It then alternates coin and step operators, e.g.

The position can progress linearly in the number of steps.

Quantum walk on a graph The walk position is any node on

the graph.

Describe the generator matrix by

The quantity is the number of edges incident on vertex .

An edge between and is denoted .

The probability distribution for a continuous walk has the differential equation

Quantum walk on a graph

Quantum mechanically we have

The natural quantum analogue is

We take

Probability is conserved because is Hermitian.

1998Farhi

Quantum walk on a graph The goal is to traverse the graph

from entrance to exit. Classically the random walk will

take exponential time. For the quantum walk, define a

superposition state

On these states the matrix elements of the Hamiltonian are

entrance exit

2002Childs, Farhi,

Gutmann

Quantum walk on a graph Add random connections

between the two trees.

All vertices (except entrance and exit) have degree 3.

Again using column states, the matrix elements of the Hamiltonian are

This is a line with a defect.

There are reflections off the defect, but the quantum walk still reaches the exit efficiently.

2003

entrance exit

Childs, Cleve, Deotto, Farhi, Gutmann,

Spielman

NAND tree quantum walk In a game tree I alternate making moves with

an opponent.

In this example, if I move first then I can always direct the ant to the sugar cube.

What is the complexity of doing this in general? Do we need to query all the leaves?

2007

AND

OR

𝑥1 𝑥2 𝑥3 𝑥4

AND AND

OR

𝑥5 𝑥6 𝑥7 𝑥8

AND

AND

Farhi, Goldstone, Gutmann

NAND tree quantum walk2007

AND

OR

𝑥1 𝑥2 𝑥3 𝑥4

AND

NAND

NAND NAND

𝑥1 𝑥2 𝑥3 𝑥4

AND

OR

𝑥1 𝑥2 𝑥3 𝑥4

AND

NOT

NOT

NOT

NOT

Farhi, Goldstone, Gutmann

NAND tree quantum walk

The Hamiltonian is a sum of an oracle Hamiltonian, representing the connections, and a fixed driving Hamiltonian, which is the remainder of the tree.

Prepare a travelling wave packet on the left. If the answer to the NAND tree problem is , then after a fixed time the

wave packet will be found on the right. The reflection depends on the solution of the NAND tree problem.

2007

wave

Farhi, Goldstone, Gutmann

Simulating quantum walks A more realistic scenario is that we have

an oracle that provides the structure of the graph; i.e., a query to a node returns all the nodes that are connected.

The quantum oracle is queried with a node number and a neighbour number .

It returns a result via the quantum operation

Here is the ’th neighbour of .

wavequery node

connected nodes

𝑈𝑂¿ 𝑥 , 𝑗⟩

¿0 ⟩

¿ 𝑥 , 𝑗⟩

¿ 𝑦 ⟩

Decomposing the Hamiltonian In the matrix picture, we have a

sparse matrix. The rows and columns correspond to

node numbers. The ones indicate connections

between nodes. The oracle gives us the position of

the ’th nonzero element in column .

𝐻=[0 0 1 0 0 1 ⋯ 00 1 0 0 0 1 ⋯ 01 0 0 0 0 0 ⋯ 10 0 0 1 1 0 ⋯ 00 0 0 1 1 0 ⋯ 01 1 0 0 0 0 ⋯ 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮0 0 1 0 0 0 ⋯ 1

]2003

Aharonov, Ta-Shma

Decomposing the Hamiltonian In the matrix picture, we have a

sparse matrix. The rows and columns correspond to

node numbers. The ones indicate connections

between nodes. The oracle gives us the position of

the ’th nonzero element in column . We want to be able to separate the

Hamiltonian into 1-sparse parts. This is equivalent to a graph

colouring – the graph edges are coloured such that each node has unique colours.

𝐻=[0 0 1 0 0 1 ⋯ 00 1 0 0 0 1 ⋯ 01 0 0 0 0 0 ⋯ 10 0 0 1 1 0 ⋯ 00 0 0 1 1 0 ⋯ 01 1 0 0 0 0 ⋯ 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮0 0 1 0 0 0 ⋯ 1

]2003

Aharonov, Ta-Shma

Graph colouring How do we do this colouring? First guess: for each node, assign

edges sequentially according to their numbering.

This does not work because the edge between nodes and may be edge (for example) of , but edge of .

Second guess: for edge between and , colour it according to the pair of numbers , where it is edge of node and edge of node .

We decide the order such that . It is still possible to have ambiguity:

say we have .

𝐻=[0 0 1 0 0 1 ⋯ 00 1 0 0 0 1 ⋯ 01 0 0 0 0 0 ⋯ 10 0 0 1 1 0 ⋯ 00 0 0 1 1 0 ⋯ 01 1 0 0 0 0 ⋯ 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮0 0 1 0 0 0 ⋯ 1

]2007

Berry, Ahokas, Cleve, Sanders

Graph colouring How do we do this colouring? First guess: for each node, assign

edges sequentially according to their numbering.

This does not work because the edge between nodes and may be edge (for example) of , but edge of .

Second guess: for edge between and , colour it according to the pair of numbers , where it is edge of node and edge of node .

We decide the order such that . It is still possible to have ambiguity:

say we have . Use a string of nodes with equal

edge colours, and compress.

𝑥

𝑦𝑧

1

11

2

2

2

3

3

3

(1,2)

(1,2)

2007Berry, Ahokas,

Cleve, Sanders

𝐻=[0 0 1 0 0 1 ⋯ 00 1 0 0 0 1 ⋯ 01 0 0 0 0 0 ⋯ 10 0 0 1 1 0 ⋯ 00 0 0 1 1 0 ⋯ 01 1 0 0 0 0 ⋯ 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮0 0 1 0 0 0 ⋯ 1

]

General Hamiltonian oracles

More generally, we can perform a colouring on a graph with matrix elements of arbitrary (Hermitian) values.

Then we also require an oracle to give us the values of the matrix elements.

𝑈𝑂¿ 𝑥 , 𝑗⟩

¿0 ⟩

¿ 𝑥 , 𝑗⟩

¿ 𝑦 ⟩

𝐻=[0 0 2 0 0 √2 𝑖 ⋯ 00 3 0 0 0 1/2 ⋯ 02 0 0 0 0 0 ⋯ −√3+ 𝑖0 0 0 1 𝑒𝑖𝜋 /7 0 ⋯ 00 0 0 𝑒−𝑖 𝜋 /7 2 0 ⋯ 0

−√2𝑖 1/2 0 0 0 0 ⋯ 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮0 0 −√3− 𝑖 0 0 0 ⋯ 1 /10

]𝑈 𝐻

¿ 𝑥 , 𝑦 ⟩

¿0 ⟩

¿ 𝑥 , 𝑦 ⟩

¿𝐻 𝑥 ,𝑦 ⟩

2003Aharonov, Ta-Shma

Simulating 1-sparse case

Assume we have a 1-sparse matrix. How can we simulate evolution under this Hamiltonian? Two cases:

1. If the element is on the diagonal, then we have a 1D subspace.

2. If the element is off the diagonal, then we need a 2D subspace.

𝐻=[0 0 0 0 0 √2𝑖 ⋯ 00 3 0 0 0 0 ⋯ 00 0 0 0 0 0 ⋯ −√3+𝑖0 0 0 1 0 0 ⋯ 00 0 0 0 2 0 ⋯ 0

−√2𝑖 0 0 0 0 0 ⋯ 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮0 0 −√3−𝑖 0 0 0 ⋯ 0

]2003

Aharonov, Ta-Shma

Simulating 1-sparse case We are given a column number . There are then 5 quantities

that we want to calculate:

1. : A bit registering whether the element is on or off the diagonal; i.e. belongs to a 1D or 2D subspace.

2. : The minimum number out of the (1D or 2D) subspace to which belongs.

3. : The maximum number out of the subspace to which belongs.

4. : The entries of in the subspace to which belongs.

5. : The evolution under for time in the subspace.

We have a unitary operation that maps

2003Aharonov, Ta-Shma

Simulating 1-sparse case

We have a unitary operation that maps

We consider a superposition of the two states in the subspace,

Then we obtain

A second operation implements the controlled operation based on the stored approximation of the unitary operation :

This gives us

Inverting the first operation then yields

2003Aharonov, Ta-Shma

Applications

2007: Discrete query NAND algorithm – Childs, Cleve, Jordan, Yeung

2009: Solving linear systems – Harrow, Hassidim, Lloyd

2009: Implementing sparse unitaries – Jordan, Wocjan

2010: Solving linear differential equations – Berry

2013: Algorithm for scattering cross section – Clader, Jacobs, Sprouse

Implementing unitaries

Construct a Hamiltonian from unitary as

Now simulate evolution under this Hamiltonian

Simulating for time gives

2009Jordan, Wocjan

Quantum simulation via walks Three ingredients:

1. A Szegedy quantum walk2. Coherent phase estimation3. Controlled state preparation

The quantum walk has eigenvalues and eigenvectors related to those for Hamiltonian.

By using phase estimation, we can estimate the eigenvalue, then implement that actually needed.

Szegedy Quantum Walk The walk uses two reflections

The first is controlled by the first register and acts on the second register.

Given some matrix , the operator is defined by

2004Szegedy

Szegedy Quantum Walk The diffusion operator is controlled by the second

register and acts on the first. Use a similar definition with matrix .

Both are controlled reflections:

The eigenvalues and eigenvectors of the step of the quantum walk

are related to those of a matrix formed from and .

2004Szegedy

Szegedy walk for simulation Use symmetric system, with

Then eigenvalues and eigenvectors are related to those of Hamiltonian.

In reality we need to modify to “lazy” quantum walk, with

Grover preparation gives

2012Berry, Childs

Szegedy walk for simulation Three step process:

1. Start with state in one of the subsystems, and perform controlled state preparation.2. Perform steps of quantum walk to approximate Hamiltonian evolution.3. Invert controlled state preparation, so final state is in one of the subsystems.

Step 2 can just be performed with small for lazy quantum walk, or can use phase estimation.

A Hamiltonian has eigenvalues , so evolution under the Hamiltonian has eigenvalues

is the step of a quantum walk, and has eigenvalues

The complexity is the maximum of

2012Berry, Childs