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

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)

Overview • Introduction to Quantum Computing • Error Correction • Memory Hierarchy • Future Work NSC '02 D. Copsey -- QC 2

Quantum Bits (qubit) • 1 qubit probabilistically represents 2 states |a C 0 |0 C 1 |1 • Additional qubits double the number of states: |ab C 00 |00 C 01 |01 C 10 |10 C 11 |11 • Quantum parallelism on an exponential number of states • Measurement collapses qubit waveform to a single classical value NSC '02 D. Copsey -- QC 3

Quantum Gates α X Gate 0 1 β α = 0 + 1 X β Bit-flip, Not 1 0 α Z Gate 1 0 α β = 0 - 1 Z β Phase-flip 0 -1 α α+β α−β H Gate 1 1 1 0 + 1 = H β Hadamard 2 1 -1 2 α T Gate 1 0 α i /4 β = e π 0 + 1 T π β e π Rotate /8 0 i /4 1 0 0 0 a Controlled Not a 00 + b 01 + 0 1 0 0 b Controlled X = 0 0 0 1 c d 10 + c 11 CNot X 0 0 1 0 d NSC '02 D. Copsey -- QC 4

CAT State Creation H 0 + 1 00 + 11 00 0 2 2 NSC '02 D. Copsey -- QC 5

Quantum Algorithms • Unordered Search: O(n 1/2 ) vs. O(n) [Grover96] • Large Number Factorization [Shor94] – O(n 3 ) vs. O(2 n/2 ) for known classical alg’s – Quantum Fourier Transform – Periodicity of Modular Exponentiation • Quantum Encryption • Quantum Teleportation NSC '02 D. Copsey -- QC 6

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] NSC '02 D. Copsey -- QC 7

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: 5x10 11 ops) • hence, error correction NSC '02 D. Copsey -- QC 8

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 NSC '02 D. Copsey -- QC 9

Quantum Error Correction Three Qubit Code Z 01 Z 12 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 NSC '02 D. Copsey -- QC 10

Error-Syndrome Measurement 0 H H Z 12 Ψ Ψ ' X 2 2 Ψ Ψ ' X 1 1 NSC '02 D. Copsey -- QC 11

3-bit Error Correction A H H Z 1 01 A H H Z 0 12 Ψ Ψ ' X Z 2 2 Ψ Ψ ' X X Z 1 1 Ψ Ψ ' X Z 0 0 NSC '02 D. Copsey -- QC 12

Concatenated Codes Logical qubit First level . . . of encoding Second level . . . . . . . . . of encoding NSC '02 D. Copsey -- QC 13

Error Correction Overhead 7-qubit code [Steane96] , applied recursively Recursion Storage Operations Min. time ( 153 k ) ( 5 k ) (7 k ) (k) 0 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 NSC '02 D. Copsey -- QC 14

Recursion Requirements from: [Oskin, et al, 2002] Shor’s Grover’s NSC '02 D. Copsey -- QC 15

Memory Hierarchy Processor Cache Memory qubits lines pages teleport teleport More physical qubits Greater density Less complex More complex operations code NSC '02 D. Copsey -- QC 16

Teleporting Between Codes source H |a |b CNOT EPR Pair target ( CAT ) |c |a X Z • Source generates |bc EPR pair • Pre-communicate |c to target with retry • Classical communication to set value • Can be used to convert between codes NSC '02 D. Copsey -- QC 17

Denser Error Correction Codes • [[7,1]] code [Steane96] – Efficient operations • [[5,1]] code [Laflamme, et al, 96] • [[8,3]] code [Steane96, Gottesman 96] Overhead per qubit Code Storage Operations 3.5 x 10 6 [[7x7x7,1]] 343 3.1 x 10 6 [[5x7x7,1]] 245 2.2 x 10 6 [[8x7x7,3]] 130.67 0.8 x 10 6 [[8x8x8,3x3x3]] < 19 NSC '02 D. Copsey -- QC 18

Quantum Fourier Transform Generic 9-qubit QFT H R R R R R R R R 2 3 4 5 6 7 8 9 H R R R R R R R 2 3 4 5 6 7 8 H R R R R R R 2 3 4 5 6 7 H R R 3 R R 5 R 6 2 4 H R R R R 5 2 3 4 H R 2 R 3 R 4 H R 2 R 3 H R 2 H Blocked for 3-qubit accesses H R R R R R R 7 R R 9 2 3 4 5 6 8 H R R R R R 6 R 7 R 8 2 3 4 5 H R R R R 5 R 6 R 7 2 3 4 H R R R R R 2 3 4 5 6 H R R 3 R 4 R 5 2 H R 2 R 3 R 4 H R R 2 3 H R 2 H NSC '02 D. Copsey -- QC 19

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 NSC '02 D. Copsey -- QC 20

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 NSC '02 D. Copsey -- QC 21