Resources for Measurement-Based Quantum Computation Quantum Carry-Lookahead

Adder

Agung Trisetyarso

Keio University

Presented in October 21st, 2008

Abstract

• Presenting the design of quantum carry-lookahead adder using measurement-based quantum computationQCLA utilizes MBQC`s ability to transfer quantum states in unit time to accelerate additionQCLA is faster than a quantum ripple-carry adder; QCLA has logarithmic depth while ripple adders have linear depth QCLA requires a cluster state that is an order of magnitude larger“Bend a Network” method results ≈26 % spatial resources optimation for in-place MBQC QCLA circuit

Content• Introduction of Raussendorf Theorem in Cluster State• Introduction of Quantum Carry-Lookahead Adder

– In-place circuit– Out-of-place circuit

• Quantum Adders: 1. Quantum Ripple-Carry Adders2. Quantum Carry-Lookahead Adder

• Performance of Measurement-Based Quantum Carry-Lookahead Adder Circuit

• Conclusions• Next Research Proposal

– Spin Cluster Qubits in All-Silicon Quantum Computer– Resources for Silica-on-Silicon Waveguide MBQC QCLA with Photon – MBQC Circuit with Fault-Tolerant

Raussendorf Theorem in Cluster State

• Quantum computation in simplest (abstract) system – Linear Transformation

– No Teleportation– No Measurement-

Driven

• Quantum computation in cluster state– Clifford Group

– Teleportation– Measurement-Driven

= U = U AU+

Quantum computation in simplest (abstract) system

• IDENTITY Gate • NOT Gate

0 1

1 = (NOT) 0

=

01

10

1

0

0

1

0 0

= (Identity) 0

=

10

01

0

1

0

1

0

Questions:

• How to deliver quantum information in real physical systems?

Raussendorf, Briegel and Browne’s Theorem

AQIS`08, KIAS

Raussendorf, Briegel and Browne’s Theorem

• Initial Eigenvalue Equations

• Measurement • Final Eigenvalue Equations

Properties of MBQC

Quantum computation in cluster state

• IDENTITY Gate • NOT Gate

Quantum computation in cluster state

Measurement Step

• Three qubits on Machine Cluster (CM) are measured in one time.

Measurement Step

• Three qubits on Machine Cluster (CM) are measured in one time.

Measurement Step

• 65 qubits on Machine Cluster (CM) are measured in first time.

• 7 qubits on CM are measured on second time

Quantum Adder

Quantum Adder

Ripple Carry AdderCarry-Lookahead

Adder

Ripple Carry Adder

Full Adder

Full Adder

Full Adder

a1a3a2 b1b2b3

CinC1C2Coutput

S1S2S3

•Multiplying full adders used with the carry ins and carry outs chained•The correct value of the carry bit ripples from one bit to the next.• The Depth is O(N) or Polynomial -> relatively slow, since each full adder must wait for the carry bit to be Calculated from the previous full adder

Vedral, Barenco and Ekert Adder

Circuit

CARRY

=SUM

=

=

UNCARRY

The Carry-Lookahead Addera1a3

a2 b1b2b3

S1S2S3

Full Adder

Full Adder

Full Adder

g0p0g1p1g2p2

Cin

C1C2Coutput

PG GG

•Using generating and propagating carries concepts

iiii

iiiii

iiiii

cpgC

babap

babag

.

),(

.),(

1

•The addition of two 1-digit

inputs ai and bi is said to generate if the addition will always carry, regardless of whether there is an input carry

•The addition of two 1-digit

inputs ai and bi is said to propagate if the addition will carry whenever there is an input carry

1 ii ba

ii ba

Implementation of CLA into Quantum Circuit

Quantum Carry-Lookahead

Adder

Addition CircuitCarry Computation

Circuit (3 procedures)

Out-of-place(5 procedures)

In-place(10 procedures)

Carry Computation Circuit• Procedures to determine the rounds:

Where:

n = logical qubits

t = sequences of rounds

m = number of rounds

Out-of-place Quantum Carry-Lookahead Adder

• The circuit aims to perform

by the following procedures:

Out-of-place QCLA circuit

• Red : G-Rounds• Blue: P-Rounds• Green: C-Rounds• Black: SUM-

Blocks

Performances and Requirements of Out-of-place MBQC QCLA

• The in-place circuit aims to erasure every unnecessary subregisters output. • The additional circuit is that it should perform:

In-place Quantum Carry-Lookahead Adder

• The implementation in Quantum circuit is expressed in following procedures:

In-place Quantum Carry-Lookahead Adder

In-place QCLA circuit

• Red : G-Rounds

• Blue: P-Rounds

• Green: C-Rounds

• Black: SUM-Blocks

Bend a Network Circuit• One may imagine the logical qubits astraveling through pipes on a two-dimensional surface.

• Horizontal and vertical axes both represent spatial axes, not temporal.

MBQC Form of

VBE

Robert Raussendorf, Daniel E. Browne, and Hans J. Briegel, Phys. Rev. A 68, 022312 (2003)

• Optimized for space, but still linear depth

•spatial resources =

304 n

“Bend a Network”In-place QCLA

Circuit

• Reduce the horizontal resources, but spent more vertically

Out-of-place MBQC QCLA circuit

• For n=10, consist of:• 4 addition circuits• 9 carry networks (2 Propagate, 3 Generate, 2 Inverse Propagate

and 2 Carry networks )

Performances and Requirements of

Out-of-place MBQC QCLA Circuit

Total Resources Qubits

Example for n=10 => Total Qubits or in-place circuit = 14657

In-place MBQC QCLA circuit

• For n=10, consist of:• 8 addition circuits• 18 carry networks (4 Propagate, 6 Generate, 4 Inverse Propagate and 4

Carry networks )

Size Comparison of Out-of-place, In-place MBQC and MBQC VBE

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o oo o oo oo o o o ooo o ooo o oooo o oo o o ooo o o o oo o ooo o o o oo oo oo

o o o o

20 40 60 80 100n

50 000

100 000

150 000

200 000

250 000

S ize

op timized mbqc

o ou t of p lace mbqcqcla

mbqcvbe

in p lace mbqcqcla

o

oo

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20 40 60 80 100n

5

10

20

50

100

200

D ep th

Depth Comparison of Out-of-place, In-place MBQC and MBQC VBE

Optimized-in-place MBQC QCLA circuit

• Diamond-like form circuit, spatial resources optimation ≈ 26 % from in-place MBQC QCLA circuit.

Optimation of MBQC QCLA Circuit

Example for n=10 => Removed Qubits/Total Qubits = 3822/14657 ≈ 26 %

Or, ≈

Conclusion

• The resources to perform quantum carry-lookahead adder in cluster state = (logical qubits, width and number of qubits in quantum gates)

• “Bend a Network” changes Manhattan grid form to Diamond-like form in MBQC QCLA circuit.

• Optimation ≈ 26 % spatial resources

Future Works(1):

• “Resources for Photonic Cluster State Computation Quantum Carry-Lookahead Adder Circuit”

References:

1. Devitt et al., Topological Cluster State with Photons, quant-ph ...

2. Stephens et. al, Deterministic optical quantum computer using photonic modules, quant-ph ...

3. Politi et. al, Silica-on-Silicon Waveguide Quantum Circuits, Science 320, 646 (2008)

• “Resources for Quantum-dot cluster state computing Quantum Carry-Lookahead Adder”

References:

1. Weinstein et. al, Quantum-dot cluster-state computing with encoded qubits, PRA 72, 020304(R) (2005)

2. Meier et. al, Quantum Computing with Spin Cluster Qubits, PRL (2003)

3. Meier et. al, Quantum Computing with antiferromagnetic spin clusters, PRB 68, 134417 (2003)

4. Skinner et. al, Hydrogenic Spin Quantum Computing in Silicon: A Digital Approach, PRL 2003

5. J.Levy, Universal Quantum Computation with Spin-1/2 Pairs and Heissenberg Exchange, PRL 2002

6. Rahman et. al, High Precision Quantum Control of Single Donor Spins in Silicon, PRL 2007.

Future Works (2):

Thank you very much

ありがとう　ございます　！！