+ Dean Copsey University of California at Davis With Mark Oskin - - PowerPoint PPT Presentation

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Memory Hierarchies for Quantum Computation

Dean Copsey

University of California at Davis

With Mark Oskin (UW), Fred Chong (Davis), Isaac Chuang (MIT), and Khaled Abdel-Gaffar (Davis)

+

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Overview

• Introduction to Quantum Computing
• Error Correction
• Memory Hierarchy
• Future Work

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Quantum Bits (qubit)

• 1 qubit probabilistically represents 2 states

|a C0|0 C1|1

• Additional qubits double the number of

states:

|ab C00|00 C01|01 C10|10 C11|11

• Quantum parallelism on an exponential

number of states

• Measurement collapses qubit waveform to a

single classical value

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Quantum Gates

1 0 - 1 X Gate Bit-flip, Not Z Gate Phase-flip = = + 2 1 2 1 H Gate Hadamard = + 01 00 + + 10 11 + Controlled Not Controlled X CNot = = T Gate Rotate /8 1 +

X Z

α β α β β α α β 1 1

• 1

1

H

α β α−β

• 1

1 1 1 α+β

T

α β

X

b a d c 1 1 1 1 a b c d e π

i /4

e π

i /4β

1 π α

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CAT State Creation

00 2 + 00 11 1 + 2

H

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Quantum Algorithms

• Unordered Search: O(n1/2) vs. O(n) [Grover96]
• Large Number Factorization [Shor94]

– O(n3) vs. O(2n/2) for known classical alg’s – Quantum Fourier Transform – Periodicity of Modular Exponentiation

• Quantum Encryption
• Quantum Teleportation

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Science Fiction?

• 5 and 7-bit machines have been built

[Vandersypen00, Laflamme99]

– NMR, ion-trap and other technologies – Shor’s and Grover’s algorithms demonstrated

• Larger machines are proposed
• Solid-state technologies are coming

[Kane98,Vrijen99,Nakamura99,Mooij99]

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General Purpose Machines

will require:

• thousands or millions of qubits
• better technology
• practical error rates are 10-6 to 10-9
• billions or trillions of operations

(e.g., factoring a 1024-bit number: 5x1011 ops)

• hence, error correction

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Quantum Error Correction

• Based on classical “linear” codes
• Some codes have efficient operations

– Steane’s [[7,1]] code

• Operate on encoded data, error correct after

each operation

• Arbitrary level of encoding for reliability
• Need to be reliable enough to run program

to completion

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Quantum Error Correction

Three Qubit Code Z01 Z12 Error Type Action +1 +1 no error no action

• 1

+1 bit 0 flipped flip bit 0

• 1
• 1

bit 1 flipped flip bit 1 +1

• 1

bit 2 flipped flip bit 2

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Error-Syndrome Measurement

H H

1

Ψ Ψ

2

Ψ Ψ Z X X

1 ' 2 ' 12

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3-bit Error Correction

H H H H Z Z Z Ψ Ψ Ψ

1

A A

2

Ψ

1

Ψ Ψ

12

Z

01

Z X X X X

2 ' 1 ' '

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Concatenated Codes

Logical qubit . . . . . . . . .

• f encoding

First level . . . Second level

• f encoding

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Error Correction Overhead

Recursion Storage Operations

• Min. time

(k) (7k) ( 153k ) ( 5k ) 1 1 1 1 7 153 5 2 49 23,409 25 3 343 3,581,577 125 4 2,401 547,981,281 625 5 16,807 83,841,135,993 3125

7-qubit code [Steane96], applied recursively

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Recursion Requirements

Shor’s Grover’s

from: [Oskin, et al, 2002]

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More physical qubits

• perations

Less complex More complex code Greater density

teleport

qubits

Cache Memory

lines pages

Processor

teleport

Memory Hierarchy

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Teleporting Between Codes

• Source generates |bc

EPR pair

• Pre-communicate |c

to target with retry

• Classical communication to set value
• Can be used to convert between codes

|a |c H source target |a |b Z X EPR Pair (CAT)

CNOT

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Denser Error Correction Codes

Overhead per qubit Code Storage Operations [[7x7x7,1]] 343 3.5 x 106 [[5x7x7,1]] 245 3.1 x 106 [[8x7x7,3]] 130.67 2.2 x 106 [[8x8x8,3x3x3]] < 19 0.8 x 106

• [[7,1]] code [Steane96] – Efficient operations
• [[5,1]] code [Laflamme, et al, 96]
• [[8,3]] code [Steane96, Gottesman 96]

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Quantum Fourier Transform

H

4 9

H R R2

8 6

R R 3

7

R R R4 H

7 6

H R H

5

R2 R R

4

R 3 R

5 3

H R R2

2

R R H

4

R

3

R

2

R H

6

R

8 4

R R5

7

R R H

5

R3 R

3 4

R

2

R R

3

R6 R

2 2

R5 R R R H

8 5

R 9 H R R R H

5 2

R7

6

R H R8 R

4

H R 6

6

R R R5

5 3

R R7

4

R R R 6 H

2

R

3

R 3

2

H R R4 H R

2

R5 R

2 2

R2 R

3

R R3

4

R R R4

3

H

4

R

3

R

2

R R7 R

Blocked for 3-qubit accesses Generic 9-qubit QFT

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Conclusion

• Classical memory hierarchies reduce cost by

accessing frequently used data fast

• Quantum memory hierarchies reduce cost by

decreasing the overhead of error correction

• Our proposal, in a nutshell

– Calculate on sparse encoding – Use denser codes for cache and RAM – Teleport between codes

• Allows for solution of larger problems

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Future Work

• Investigate different physical media

– Different technologies, different primitives – Decoherence-free subspaces – High-genus surfaces

• Investigate larger “linear” codes
• Investigate non-linear codes

– Toric codes – Iteratively decoded codes