``hamiltonian quantum cellular automata in 1d'' by daniel...
TRANSCRIPT
![Page 1: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/1.jpg)
“Hamiltonian Quantum Cellular Automata in 1D”by Daniel Nagaj and Pawel Wocjan
Maris Ozols
University of WaterlooDepartment of C&O
April 3, 2008
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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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Introduction
![Page 4: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/4.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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. . .
![Page 5: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/5.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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. . .
![Page 6: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/6.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
![Page 7: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/7.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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.
![Page 8: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/8.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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.
![Page 9: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/9.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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.
![Page 10: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/10.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Main ingredients
1. Universal set of quantum gates
2. Classical cellular automaton
3. Quantum walk (for required time analysis)
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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.
![Page 12: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/12.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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.
![Page 13: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/13.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
![Page 14: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/14.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
![Page 15: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/15.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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.
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Circuits in “ladder form”
Original circuit
q1
q2
q3
W
S W
Modified circuit
q1
q2
q3
W
S
I
W
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
The first construction (d = 10)
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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}.
![Page 19: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/19.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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 0
input
Basis states are marked by elements of {�,I,W, S, I} × {0, 1}.
![Page 20: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/20.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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}.
![Page 21: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/21.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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}.
![Page 22: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/22.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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 Wdata 0 0 0 0 0 0 q1 q2 q3 0 0 0
input
Basis states are marked by elements of {�,I,W, S, I} × {0, 1}.
![Page 23: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/23.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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 Wdata 0 0 0 0 0 0 q1 q2 q3 0 0 0
input
Basis states are marked by elements of {�,I,W, S, I} × {0, 1}.
![Page 24: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/24.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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 Wdata 0 0 0 0 0 0 q1 q2 q3 0 0 0
input
Basis states are marked by elements of {�,I,W, S, I} × {0, 1}.
![Page 25: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/25.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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 Wdata 0 0 0 0 0 0 q1 q2 q3 0 0 0
input
Basis states are marked by elements of {�,I,W, S, I} × {0, 1}.
![Page 26: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/26.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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 Wdata 0 0 0 0 0 0 q1 q2 q3 0 0 0
input
Basis states are marked by elements of {�,I,W, S, I} × {0, 1}.
![Page 27: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/27.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
![Page 28: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/28.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
![Page 29: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/29.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I � � I W I S I I W0 0 0 0 0 0 | q12 〉 q3 0 0 0
![Page 30: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/30.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I � � I W S I I I W0 0 0 0 0 0 | q123 〉 0 0 0
![Page 31: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/31.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I � � I W S I I I W0 0 0 0 0 0 | q123 〉 0 0 0
![Page 32: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/32.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I � � I W S I I I W0 0 0 0 0 0 | q123 〉 0 0 0
![Page 33: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/33.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I � � I W S I I W I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 34: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/34.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I � I � W S I I W I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 35: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/35.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I � I W � S I I W I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 36: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/36.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I I � W � S I I W I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 37: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/37.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I I � W � S I I W I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 38: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/38.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I I � W S � I I W I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 39: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/39.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I I � W S I � I W I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 40: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/40.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I I � W S I I � W I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 41: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/41.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I I � W S I I W � I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 42: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/42.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I I W � S I I W � I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 43: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/43.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I I W S � I I W � I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 44: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/44.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I I W S I � I W � I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 45: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/45.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I I W S I I � W � I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 46: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/46.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I I W S I I W � � I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 47: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/47.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I W I S I I W � � I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 48: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/48.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I W S I I I W � � I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 49: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/49.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I W S I I I W � � I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 50: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/50.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I W S I I I W � � I0 0 0 0 0 0 | q123 〉 0 0 0
![Page 51: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/51.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� � I W S I I W I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 52: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/52.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� I � W S I I W I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 53: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/53.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� I W � S I I W I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 54: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/54.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� I W S � I I W I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 55: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/55.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� I W S I � I W I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 56: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/56.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� I W S I I � W I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 57: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/57.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
� I W S I I W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 58: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/58.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
I � W S I I W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 59: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/59.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
I W � S I I W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 60: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/60.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
I W S � I I W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 61: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/61.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
I W S I � I W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 62: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/62.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
I W S I I � W � I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 63: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/63.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration
I W S I I W � � I � � I0 0 0 0 0 0 | s123 〉 0 0 0
![Page 64: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/64.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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}
![Page 65: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/65.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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.
![Page 66: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/66.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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.
![Page 67: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/67.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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.
![Page 68: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/68.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
“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\
![Page 69: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/69.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
The second construction (d = 20)
![Page 70: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/70.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Encoding the circuit
Circuit in “ladder form”
q1
q2
q3
W
S
I
W
Encoded circuit
1st sequence 2nd sequence
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} .
![Page 71: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/71.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Encoding the circuit
Circuit in “ladder form”
q1
q2
q3
W
S
I
W
Encoded circuit
1st sequence 2nd sequence
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} .
![Page 72: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/72.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Encoding the circuit
Circuit in “ladder form”
q1
q2
q3
W
S
I
W
Encoded circuit
1st sequence 2nd sequence
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} .
![Page 73: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/73.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Encoding the circuit
Circuit in “ladder form”
q1
q2
q3
W
S
I
W
Encoded circuit
1st sequence 2nd sequence
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} .
![Page 74: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/74.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Encoding the circuit
Circuit in “ladder form”
q1
q2
q3
W
S
I
W
Encoded circuit1st sequence
2nd sequence
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} .
![Page 75: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/75.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Encoding the circuit
Circuit in “ladder form”
q1
q2
q3
W
S
I
W
Encoded circuit1st sequence 2nd sequence
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} .
![Page 76: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/76.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Encoding the circuit
Circuit in “ladder form”
q1
q2
q3
W
S
I
W
Encoded circuit1st sequence 2nd sequence
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} .
![Page 77: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/77.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Encoding the circuit
Circuit in “ladder form”
q1
q2
q3
W
S
I
W
Encoded circuit1st sequence 2nd sequence
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
Basis states are marked by elements of
{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Encoding the circuit
Circuit in “ladder form”
q1
q2
q3
W
S
I
W
Encoded circuit1st sequence 2nd sequence
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
Basis states are marked by elements of
{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Encoding the circuit
Circuit in “ladder form”
q1
q2
q3
W
S
I
W
Encoded circuit1st sequence 2nd sequence
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
Basis states are marked by elements of
{W,S, I, W©, S©, I©,I,B,, �} × {0, 1} .
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
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Current configuration� � � � � I W S I I I W �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
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Current configuration� � � � � I W S I I I W© � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
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Current configuration� � � � � I W S I I I© W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
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Current configuration� � � � � I W S I I© I W � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
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q3
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
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q3
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
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Current configuration� � I W S I I I W B � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
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Current configuration� � I W S I I I W B � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
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Current configuration� � I W S I I I W � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
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Current configuration� � I W S I I I W � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
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Current configuration� � I W S I I I W© � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
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Current configuration� � I W S I I I© W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
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Current configuration� � I W S I I© I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
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Current configuration� � I W S I© I I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
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Current configuration� � I W S© I I I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
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Current configuration� � I W© S I I I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
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Current configuration� � I© W S I I I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
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Current configuration� I W S I I I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
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Current configuration� I W S I I I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
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Current configuration� I W I S I I I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
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Current configuration� I W S I I I I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
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Current configuration� I W S I I I I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
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W
S
I
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Current configuration� I W S I I I I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
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W
S
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Current configuration� I W S I I I I W � � � � �0 1 0 0 0 1 | q123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
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W
S
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Current configuration� I W S I I I W I � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
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W
S
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Current configuration� I W S I I I W I � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
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W
S
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Current configuration� I W S I I I W � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
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q3
W
S
I
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Current configuration� I W S I I I W � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
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Current configuration� I W S I I I W© � � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
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Current configuration� I W S I I I© W � � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration� I W S I I© I W � � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration� I W S I© I I W � � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration� I W S© I I I W � � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
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Current configuration� I W© S I I I W � � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration� I© W S I I I W � � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Evolution
Circuit
q1
q2
q3
W
S
I
W
Current configuration I W S I I I W � � � � � �0 1 0 0 0 1 | s123 〉 1 0 0 0 1
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Rules 1
Moving the active spot
A → A© �
A B© → A© B
� A© → A
Turning around
�
1→ � I
1
�
0→ � B
0
![Page 195: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/195.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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.
![Page 197: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/197.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
“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\
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Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
Conclusion
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Introduction The first construction (d = 10) The second construction (d = 20) 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)
![Page 200: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/200.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) 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)
![Page 201: ``Hamiltonian Quantum Cellular Automata in 1D'' by Daniel ...home.lu.lv/~sd20008/papers/essays/Hamiltonian Quantum Cellular A… · Hamiltonian Quantum cellular automaton How it works?](https://reader030.vdocument.in/reader030/viewer/2022040620/5f31b0add5828e55f563250c/html5/thumbnails/201.jpg)
Introduction The first construction (d = 10) The second construction (d = 20) Conclusion
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/
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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