Quantum Computing Architectures 1:-2: Fred Chong (UCD) - Intro, quantum algorithms, and error correction 2:-2:3 Break and discussion 2:3-3:3 Ike Chuang (MIT) - Device technology and implementation issues 3:3-4: Break and official refreshments 4:-5: Mark Oskin (UW) - Quantum architectures 5:- Discussion Plenty of time for questions and discussion All materials available at: http://www.cs.washington.edu/homes/oskin/quantum-tutorial ISCA 5/2 F. Chong -- QC 1
Quantum Computing for Architects + Fred Chong University of California at Davis
Science Fiction? 5 and 7-bit machines have been built 1-bit machines are planned Better technologies are coming Why architectural study? [Vandersypen1, Laflamme99] [Kane98,Vrijen99,Nakamura99,Mooij99] perspective to guide device development ISCA 5/2 F. Chong -- QC 3
Outline Quantum bits and operations Algorithms Quantum search Factorization Error correction Teleportation ISCA 5/2 F. Chong -- QC 4
Quantum Bits (qubit) + 1 qubit probabilistically represents 2 states a> = C > + C 1 1> Every additional qubit doubles # states ab> = C > + C 1 1> + C 1 1> +C 11 11> Quantum parallelism on an exponential number of states But measurement collapses qubits to single classical values ISCA 5/2 F. Chong -- QC 5
7 qubit Quantum Computer ( Vandersypen, Steffen, Breyta, Yannoni, Sherwood, and Chuang, 21 ) Bulk spin NM: nuclear spin qubits Decoherence in 1 sec; operations at 1 Kz Failure probability = 1-3 per operation Potentially 1 sec @ 1 Kz = 1-6 per op pentafluorobutadienyl cyclopentadienyldicarbonyliron complex ISCA 5/2 F. Chong -- QC 6
Silicon Technology ( Kane, Nature 393, p133, 1998 ) ISCA 5/2 F. Chong -- QC 7
Quantum Operations Manipulate probability amplitudes Must conserve energy Must be reversible ISCA 5/2 F. Chong -- QC 8
ISCA 5/2 F. Chong -- QC 9 Bit Flip Flips probabilities for > and 1> Conservation of energy eversibility => unitary matrix (* means complex conjugate) 1 X Gate Bit-flip, Not = + X α β α β 1 1 1 2 2 2 = + = β α C i i I X X T = = 1 1 1 1 ) ( *
Bloch Sphere Visualize a qubit as a vector on a sphere z f Y Operations composed of a rotation primitive x q y 1 ISCA 5/2 F. Chong -- QC 1
Controlled Not Controlled Not Controlled X CNot X 1 1 1 1 a b c d = a + b1 + d1 + c11 Control bit determines whether X operates Control bit is affected by operation ISCA 5/2 F. Chong -- QC 11
Quantum subsumes Classical A B C A B C AB Toffoli gate, or controlled-controlled-not NAND does not conserve energy Number of inputs must equal number of outputs Toffoli gate simulates NAND Inputs = a,b; c set to 1 Output = c ISCA 5/2 F. Chong -- QC 12
Universal Quantum Operations Gate adamard 1 1 1 α (α + β) > + (α β) 1 > = 2 1 1 β 2 T Gate T α - iπ = αe 8 β e iπ - iπ iπ 8 e 8 8 > + βe 1 > Z Gate Phase-flip Z 1 α -1 β = α > β 1 > Controlled Not Controlled X CNot X 1 1 1 a b 1 c d = a > + b 1 > + d 1 > + e 11 > ISCA 5/2 F. Chong -- QC 13
Quantum Algorithms Search (function evaluation) Factorization (FFT, discrete log) Key distribution Digital signatures Clock synchronization ISCA 5/2 F. Chong -- QC 14
Quantum Parallelism > + 1 > 2 x x Uf ψ > > y y f(x) f(x) :{,1} {,1} ψ > =, f() > + 1, f(1) > 2 ISCA 5/2 F. Chong -- QC 15
Deutch s Algorithm(1) [Deutch 85] > x x Uf 1> y y f(x) ψ 1 > ψ 1 >= > + 1 > > 1 > 2 2 ISCA 5/2 F. Chong -- QC 16
Deutch s Algorithm(2) > x x Uf 1> y y f(x) Note that the xor just flips the probabilities for > and 1> : U f > 1 > f x) x > = ( 1) x 2 > 1 > 2 ( > ISCA 5/2 F. Chong -- QC 17
Deutch s Algorithm(3) > x x Uf 1> y y f(x) ψ 2 > ψ 2 > = > + 1> > 1> ± 2 2 > 1> > 1> ± 2 2 if if f ( ) = f (1) f ( ) f (1) ISCA 5/2 F. Chong -- QC 18
Deutch s Algorithm(4) > x x Uf 1> y y f(x) ψ 3 > = > 1> ± > 2 > 1> ± 1> 2 if if ψ 3 > f ( ) = f (1) f ( ) f (1) = ± f () f (1) > > 1 > 2 ISCA 5/2 F. Chong -- QC 19
Quantum Factorization For N = pq, where p,q are large primes, find p,q given N Let r = Order(x,N), which is min value > such that x r mod N = 1, x coprime N Then (x r/2 +/- 1) mod N = p,q eg Order(2, 15) = 4 (x 4/2 +/-1) mod 15 = 3,5 ISCA 5/2 F. Chong -- QC 2
Shor s Algorithm [Shor94] m qubits > r r FT s/r n qubits > x r mod N j j=1: r> = > + 4> + 8> + 12> + j=2: r> = 1> + 5> + 9> + 13> + j=4: r> = 2> + 6> + 1> + 14> + j=8: r> = 3> + 7> + 11> + 15> + ISCA 5/2 F. Chong -- QC 21
Quantum Fourier Transform r is in the period, but how to measure r? QFT takes period r => period s/r Measurement yields I*s/r for some I educe fraction I*s/r => r is the denominator with high probability! epeat algorithm if pq not equal N O(n 3 ) instead of O(2 n )!!! ISCA 5/2 F. Chong -- QC 22
Quantum Search (function evaluation) n qubits > G G G measure Oracle workspace Iteratively concentrates probability towards desired measurement [Grover96] Can search N unordered items in N time N ISCA 5/2 F. Chong -- QC 23
Error Correction is Crucial Need continuous error correction can operate on encoded data [Shor96, Steane96, Gottesman99] Threshold Theorem [Ahanorov 97] failure rate of 1-4 per op can be tolerated Practical error rates are 1-6 to 1-9 ISCA 5/2 F. Chong -- QC 24
Quantum Error Correction Z 12 Z 23 Error Type Action +1 +1 no error no action +1-1 bit 3 flipped flip bit 3-1 +1 bit 1 flipped flip bit 1-1 -1 bit 2 flipped flip bit 2 (3-qubit code) ISCA 5/2 F. Chong -- QC 25
Syndrome Measurment X X Z 12 Ψ2 Ψ 2 ' Ψ1 Ψ 1 ' ISCA 5/2 F. Chong -- QC 26
3-bit Error Correction A 1 X X Z 1 A X X Z 12 Ψ 2 X Ψ 2 ' Ψ 1 X Ψ 1 ' Ψ X Ψ ' ISCA 5/2 F. Chong -- QC 27
Concatenated Codes Logical qubit First level of encoding... Second level of encoding......... ISCA 5/2 F. Chong -- QC 28
Error Correction Overhead 7-qubit code [Steane96], applied recursively ecursion Storage Operations Min. time (k) (7 k ) ( 153 k ) ( 5 k ) 1 1 1 1 7 153 5 2 49 23,49 25 3 343 3,581,577 125 4 2,41 547,981,281 625 5 16,87 83,841,135,993 3125 ISCA 5/2 F. Chong -- QC 29
ecursion equirements Shor s Grover s ISCA 5/2 F. Chong -- QC 3
Clustering ecursive scheme is overkill Don t error correct every operation [Oskin,Chong,Chuang IEEE Computer 2] ISCA 5/2 F. Chong -- QC 31
Space Savings Shor s p=1-6 Grover s ISCA 5/2 F. Chong -- QC 32 p=1-6
Time Savings Shor s p=1-6 Grover s ISCA 5/2 F. Chong -- QC 33 p=1-6
Teleportation Sender a> 1 qubit 2 classical bits eceiver a> Destroy source qubit and recreates at target Pre-communicate half of a CAT state ISCA 5/2 F. Chong -- QC 34
CAT State > > > + 1 > > 2 > + 11 > 2 Two bits are in lockstep both or both 1 Named for Shrodinger s cat Also EP pair for Einstein, Podolsky, osen ISCA 5/2 F. Chong -- QC 35
Teleportation Circuit source a> EP Pair (CAT) b> target c> CNOT X Z a> Source generates bc> EP pair Pre-communicate c> to target with retry Classical communication to set value Can be used to convert between codes! ISCA 5/2 F. Chong -- QC 36
Memory ierarchy Processor qubits Cache lines Memory pages teleport teleport [[343,1,15]] [[245,1,15]] [[392,3,15 More physical qubits Less complex operations Greater de More compl code ISCA 5/2 F. Chong -- QC 37
ISCA 5/2 F. Chong -- QC 38 Quantum Fourier Transform 4 9 2 8 6 3 7 4 7 6 5 2 4 3 5 3 2 2 4 3 2 6 8 4 5 7 5 3 3 4 2 3 6 2 2 5
Blocked QFT 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 2 3 4 5 6 7 2 3 4 5 6 2 3 4 5 2 3 4 2 3 2 ISCA 5/2 F. Chong -- QC 39
Summary Quantum computers can be built Error correction allows scalability Tremendous potential for some applications ISCA 5/2 F. Chong -- QC 4
est of the afternoon Isaac Chuang - Quantum devices ow things really work Mark Oskin Quantum architectures What architects can do ISCA 5/2 F. Chong -- QC 41