Quantum Computing Architectures

Size: px

Start display at page:

Download "Quantum Computing Architectures"

Alice Walker
9 years ago
Views:

1 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: ISCA 5/2 F. Chong -- QC 1

official refreshments 4:-5: Mark Oskin (UW) - Quantum architectures 5:- Discussion Plenty of time for questions

2 Quantum Computing for Architects + Fred Chong University of California at Davis

3 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

4 Outline Quantum bits and operations Algorithms Quantum search Factorization Error correction Teleportation ISCA 5/2 F. Chong -- QC 4

5 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

1> +C 11 11> Quantum parallelism on an exponential number of states But

6 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 1 Kz = 1-6 per op pentafluorobutadienyl cyclopentadienyldicarbonyliron complex ISCA 5/2 F. Chong -- QC 6

1 Kz Failure probability = 1-3 per operation Potentially 1 sec @ 1 Kz = 1-6 per op

7 Silicon Technology ( Kane, Nature 393, p133, 1998 ) ISCA 5/2 F. Chong -- QC 7

8 Quantum Operations Manipulate probability amplitudes Must conserve energy Must be reversible ISCA 5/2 F. Chong -- QC 8

9 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 = + = β α C i i I X X T = = ) ( *

10 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

11 Controlled Not Controlled Not Controlled X CNot X 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

12 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

of inputs must equal number of outputs Toffoli gate simulates

13 Universal Quantum Operations Gate adamard α (α + β) > + (α β) 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 a b 1 c d = a > + b 1 > + d 1 > + e 11 > ISCA 5/2 F. Chong -- QC 13

Gate Phase-flip Z 1 α -1 β = α > β 1 > Controlled Not Controlled X CNot

14 Quantum Algorithms Search (function evaluation) Factorization (FFT, discrete log) Key distribution Digital signatures Clock synchronization ISCA 5/2 F. Chong -- QC 14

15 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

16 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

17 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

18 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

19 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

20 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

that x r mod N = 1, x coprime N Then (x r/2 +/- 1) mod N = p,q

21 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

22 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

23 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

24 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

25 Quantum Error Correction Z 12 Z 23 Error Type Action no error no action +1-1 bit 3 flipped flip bit bit 1 flipped flip bit bit 2 flipped flip bit 2 (3-qubit code) ISCA 5/2 F. Chong -- QC 25

26 Syndrome Measurment X X Z 12 Ψ2 Ψ 2 ' Ψ1 Ψ 1 ' ISCA 5/2 F. Chong -- QC 26

27 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

28 Concatenated Codes Logical qubit First level of encoding... Second level of encoding ISCA 5/2 F. Chong -- QC 28

29 Error Correction Overhead 7-qubit code [Steane96], applied recursively ecursion Storage Operations Min. time (k) (7 k ) ( 153 k ) ( 5 k ) , ,581, ,41 547,981, ,87 83,841,135, ISCA 5/2 F. Chong -- QC 29

30 ecursion equirements Shor s Grover s ISCA 5/2 F. Chong -- QC 3

31 Clustering ecursive scheme is overkill Don t error correct every operation [Oskin,Chong,Chuang IEEE Computer 2] ISCA 5/2 F. Chong -- QC 31

32 Space Savings Shor s p=1-6 Grover s ISCA 5/2 F. Chong -- QC 32 p=1-6

33 Time Savings Shor s p=1-6 Grover s ISCA 5/2 F. Chong -- QC 33 p=1-6

34 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

35 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

36 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

37 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

38 ISCA 5/2 F. Chong -- QC 38 Quantum Fourier Transform

39 Blocked QFT ISCA 5/2 F. Chong -- QC 39

40 Summary Quantum computers can be built Error correction allows scalability Tremendous potential for some applications ISCA 5/2 F. Chong -- QC 4

41 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

Bits Superposition Quantum Parallelism

Bits Superposition Quantum Parallelism 7-Qubit Quantum Computer Typical Ion Oscillations in a Trap Bits Qubits vs Each qubit can represent both a or at the same time! This phenomenon is known as Superposition. It leads to Quantum Parallelism