``hamiltonian quantum cellular automata in 1d'' by daniel...

“Hamiltonian Quantum Cellular Automata in 1D”by Daniel Nagaj and Pawel Wocjan

Maris Ozols

University of WaterlooDepartment of C&O

April 3, 2008

Introduction The first construction (d = 10) The second construction (d = 20) Conclusion

Outline

1. Introduction

2. Universal HQCA with d = 10

3. Universal HQCA with d = 20

4. Conclusion

HQCA – Hamiltonian Quantum cellular automatonUniversal – can be used to solve all problems in BQPd – the dimension of the state space of each cell


Introduction


Hamiltonian Quantum cellular automaton

How it works?

1. Take a rotating supermassive black hole

2. Shine a certain amount of the dark energy into it

3. Measure the change of the cosmological constant with respectto the Friedmann-Lemaıtre-Robertson-Walker metric

I’m joking. . .


Hamiltonian Quantum cellular automaton

How it works?

1. Prepare a basis state containing both – the input and theprogram itself

2. Let this state evolve undisturbed according to some fixedHamiltonian for some amount of time

3. Measure a small subsystem in the computational basis to getthe answer


Motivation

For meI had to choose some topic

Two women wiring the right side of the ENIACwith a new program, in the “pre-von Neumann” days.

Standing: Ester Gerston, Crouching: Gloria Ruth Gorden.


Motivation

For meIt’s a nice idea – reminds me of von Neumann architecture,universal Turing machine and ENIAC. . .

Two women wiring the right side of the ENIACwith a new program, in the “pre-von Neumann” days.

Standing: Ester Gerston, Crouching: Gloria Ruth Gorden.


Main ingredients

1. Universal set of quantum gates

2. Classical cellular automaton

3. Quantum walk (for required time analysis)


Real universal gates

Theorem (Rudolph & Grover, 2002)

The gate

G(φ) =

1 0 0 00 1 0 00 0 cosφ − sinφ0 0 sinφ cosφ

is universal for quantum computing, when φ is an irrationalmultiple of π. [RG02]Idea: Simulate one-qubit gates and CNOT, which are universalaccording to [BBC+95].

NoteWe can apply G to any pair of qubits. We have to:

I simulate the real and imaginary parts separately,

I use ancilla qubits.

Thus we get universality only in some subspace.


The W gate

Claim (Nagaj & Wocjan, 2008)

The gate

W =

1 0 0 00 1 0 00 0 1√

2− 1√

2

0 0 1√2

1√2

= G(π4 )

is universal for quantum computing. [NW08]Idea: Simulate Toffoli and Hadamard gates, which are universalaccording to [Aha03]. How?

Swaping qubits

S =

1 0 0 00 0 1 00 1 0 00 0 0 1


The main challenge

We would like our Hamiltonian to be as simple as possible:

I it acts on a 1D chain of qudits,

I it is local – only nearest-neighbor interactions,

I it is translationally invariant,

I qudits have as small dimension as possible.


Circuits in “ladder form”

Original circuit

q1

q2

q3

W

S W

Modified circuit

q1

q2

q3

W

S

I

W


The first construction (d = 10)


Encoding the circuit

Circuit in “ladder form”

q1

q2

q3

W

S

I

W

Encoded circuit

2nd marker 1st marker 1st sequence 2nd sequence

program

� � I � � I I W S I I W

data

0 0 0 0 0 0 q1 q2 q3 0 0 0input

Basis states are marked by elements of {�,I,W, S, I} × {0, 1}.




q1

q2

q3

W

S

I

W

Encoded circuit


program

� � I � � I I W S I I W

data

0 0 0 0 0 0

q1 q2 q3

0 0 0

input





q1

q2

q3

W

S

I

W

Encoded circuit


program

� � I � � I I W S I I W

data 0 0 0 0 0 0 q1 q2 q3 0 0 0input





q1

q2

q3

W

S

I

W

Encoded circuit

2nd marker 1st marker

1st sequence

2nd sequence

program

� � I � � I I

W S

I I W

data 0 0 0 0 0 0 q1 q2 q3 0 0 0input





q1

q2

q3

W

S

I

W

Encoded circuit


1st sequence 2nd sequence

program

� � I � � I I

W S

I

I Wdata 0 0 0 0 0 0 q1 q2 q3 0 0 0

input





q1

q2

q3

W

S

I

W

Encoded circuit



program

� � I � � I

I W S I I Wdata 0 0 0 0 0 0 q1 q2 q3 0 0 0

input





q1

q2

q3

W

S

I

W

Encoded circuit

2nd marker

1st marker 1st sequence 2nd sequence

program

� � I

� � I I W S I I Wdata 0 0 0 0 0 0 q1 q2 q3 0 0 0

input





q1

q2

q3

W

S

I

W

Encoded circuit


program � � I � � I I W S I I Wdata 0 0 0 0 0 0 q1 q2 q3 0 0 0

input



Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration

� � I � � I I W S I I W0 0 0 0 0 0 q1 q2 q3 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I � � I W I S I I W0 0 0 0 0 0 | q12 〉 q3 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I � � I W S I I I W0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I � � I W S I I W I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I � I � W S I I W I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I � I W � S I I W I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I I � W � S I I W I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I I � W S � I I W I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I I � W S I � I W I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I I � W S I I � W I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I I � W S I I W � I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I I W � S I I W � I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I I W S � I I W � I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I I W S I � I W � I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I I W S I I � W � I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I I W S I I W � � I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I W I S I I W � � I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I W S I I I W � � I0 0 0 0 0 0 | q123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� � I W S I I W I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� I � W S I I W I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� I W � S I I W I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� I W S � I I W I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� I W S I � I W I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� I W S I I � W I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


� I W S I I W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


I � W S I I W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


I W � S I I W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


I W S � I I W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


I W S I � I W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


I W S I I � W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Evolution

Circuit

q1

q2

q3

W

S

I

W


I W S I I W � � I � � I0 0 0 0 0 0 | s123 〉 0 0 0


Rules

Move gate

� A → A � where A ∈ {W,S, I}

Apply gate

I A

x y→ A I

A(x, y)where A ∈ {W,S, I}


HamiltonianThese rules give us

H10 = −L−1∑j=1

(R+R†)(j,j+1)

R =∑

A∈{W,S,I}

(|A�〉〈�A| ⊗ I + |A I〉〈I A| ⊗A

)

Consider how H10 acts on the program register. If we map {I, �}to |0〉 (no fermion) and {W,S, I} to |1〉 (a fermion), then we get

Hfermion = −L−1∑j=1

(|10〉〈01|+ |01〉〈10|

)(j,j+1)

or diffusion of a discrete free fermion gas on a line! In other words:the initial state remains in the subspace spaned by the basisvectors of the same Hamming weight – fermions jumping around.



H10 = −L−1∑j=1

(R+R†)(j,j+1)

R =∑

A∈{W,S,I}

(|A�〉〈�A| ⊗ I + |A I〉〈I A| ⊗A

)Consider how H10 acts on the program register. If we map {I, �}to |0〉 (no fermion) and {W,S, I} to |1〉 (a fermion), then we get


(|10〉〈01|+ |01〉〈10|

)(j,j+1)

or diffusion of a discrete free fermion gas on a line!

In other words:the initial state remains in the subspace spaned by the basisvectors of the same Hamming weight – fermions jumping around.



H10 = −L−1∑j=1

(R+R†)(j,j+1)

R =∑

A∈{W,S,I}

(|A�〉〈�A| ⊗ I + |A I〉〈I A| ⊗A

)Consider how H10 acts on the program register. If we map {I, �}to |0〉 (no fermion) and {W,S, I} to |1〉 (a fermion), then we get


(|10〉〈01|+ |01〉〈10|

)(j,j+1)

or diffusion of a discrete free fermion gas on a line! In other words:the initial state remains in the subspace spaned by the basisvectors of the same Hamming weight – fermions jumping around.


“Fermion love story”

Example with 5 sites and 2 fermions. . .♥

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

È00011\È00110\ È00101\È01100\ È01010\ È01001\È11000\ È10100\ È10010\ È10001\


The second construction (d = 20)




q1

q2

q3

W

S

I

W

Encoded circuit


program

� � � � � � I W S I I I W

data

0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1boundary markers input

Basis states are marked by elements of

{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .




q1

q2

q3

W

S

I

W

Encoded circuit


program

� � � � � � I W S I I I W

data

0 1 0 0 0 1

q1 q2 q3

1 0 0 0 1boundary markers

input


{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .




q1

q2

q3

W

S

I

W

Encoded circuit


program

� � � � � � I W S I I I W

data

0

1

0 0 0

1 q1 q2 q3 1

0 0 0

1boundary markers input


{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .




q1

q2

q3

W

S

I

W

Encoded circuit


program

� � � � � � I W S I I I W

data 0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1boundary markers input


{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .




q1

q2

q3

W

S

I

W

Encoded circuit1st sequence

2nd sequence

program

� � � � � � I

W S

I I I W



{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .




q1

q2

q3

W

S

I

W

Encoded circuit1st sequence 2nd sequence

program

� � � � � � I

W S

I I

I W



{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .




q1

q2

q3

W

S

I

W


program

� � � � � �

I W S I I I W



{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .




q1

q2

q3

W

S

I

W


program

� � � � � �

I W S I I I W data 0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1

boundary markers input


{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .




q1

q2

q3

W

S

I

W


program � � � � � � I W S I I I W data 0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1

boundary markers input


{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � � I W S I I I W 0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � � I W S I I I W© �0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � � I W S I I I© W �0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � � I W S I I© I W �0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � � I W S I© I I W �0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � � I W S© I I I W �0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � � I W© S I I I W �0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � � I© W S I I I W �0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I W S I I I W �0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I I W S I I I W �0 1 0 0 0 1 q1 q2 q3 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I W I S I I I W �0 1 0 0 0 1 | q12 〉 q3 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I W S I I I I W �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I W S I I I W I �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I W S I I I W �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I W S I I I W© � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I W S I I I© W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I W S I I© I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I W S I© I I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I W S© I I I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I W© S I I I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � � I© W S I I I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W S I I I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � B I W S I I I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I B W S I I I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W B S I I I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W S B I I I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W S I B I I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W S I I B I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W S I I I B W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W S I I I W B � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W S I I I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W S I I I W© � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W S I I I© W � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W S I I© I W � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

I

W

Current configuration� � � � I W S I© I I W � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1


Evolution

Circuit

q1

q2

q3

W

S

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Rules 2

Apply gates

I A

x y→ A I

A(x, y)

I �

1→ �

1

Shift gates

B A → A B

B �

0→ �

0


Hamiltonian

These rules give us the following Hamiltonian:

H20 = −L−1∑j=1

∑r∈rules

(Rr +R†r)(j,j+1)

Since the cellular automaton is deterministic and reversible, we canchange the basis so that our Hamiltonian becomes

Hline = −T−1∑t=0

(|t〉〈t+ 1|+ |t+ 1〉〈t|

)This is a continuous time quantum walk on a line of length T + 1.


“Lonely fermion”

Example with 5 sites and 1 fermion

0 2 4 6 8 10 12

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

0.2

0.4

0.6

0.8

1.0È10000\ È01000\ È00100\ È00010\ È00001\


Conclusion


Conclusion

Hamiltonian QCA

I on a 1D chain of qudits

I only nearest-neighbor interactions

I translationally invariant rules

Evolution

I diffusion of fermion gas (d = 10)

I quantum walk on a line (d = 20)


References

Daniel Nagaj, Pawel Wocjan, Hamiltonian Quantum CellularAutomata in 1D, arXiv:0802.0886. Pawel’s talk at PI isavailable at pirsa-08010008.

Terry Rudolph, Lov Grover, A 2 rebit gate universal forquantum computing, arXiv:quant-ph/0210187.

A. Barenco, C.H. Bennett, R. Cleve, D.P. DiVincenzo, N.Margolus, P. Shor, T. Sleator, J. Smolin, H. Weinfurter,Elementary gates for quantum computation,arXiv:quant-ph/9503016.

Dorit Aharonov, A Simple Proof that Toffoli and Hadamardare Quantum Universal, arXiv:quant-ph/0301040.

Pictures are taken fromhttp://ftp.arl.army.mil/ftp/historic-computers/

http://arxiv.org/abs/0802.0886

http://pirsa.org/08010008

http://arxiv.org/abs/quant-ph/0210187



http://ftp.arl.army.mil/ftp/historic-computers/

Thank you for your attention!

Left: Patsy Simmers, holding ENIAC boardNext: Mrs. Gail Taylor, holding EDVAC board

Next: Mrs. Milly Beck, holding ORDVAC boardRight: Mrs. Norma Stec, holding BRLESC-I board

``hamiltonian quantum cellular automata in 1d'' by daniel...

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