Quantum Bits (qubit) 1 qubit probabilistically represents 2 states |a> = C0|0> + C1|1> Every additional qubit doubles # states |ab> = C00|00> + C01|01> + C10|10> +C11|11> Quantum parallelism on an exponential number of states But measurement collapses qubits to single classical values Boulder 2/03 F. Chong -- QC

7 qubit Quantum Computer ( Vandersypen, Steffen, Breyta, Yannoni, Sherwood, and Chuang, 2001 ) Bulk spin NMR: nuclear spin qubits Decoherence in 1 sec; operations at 1 KHz Failure probability = 10-3 per operation Potentially 100 sec @ 10 KHz = 10-6 per op pentafluorobutadienyl cyclopentadienyldicarbonyliron complex Boulder 2/03 F. Chong -- QC

Silicon Technology ( Kane, Nature 393, p133, 1998 ) Boulder 2/03 F. Chong -- QC

Quantum Operations Manipulate probability amplitudes Must conserve energy Must be reversible Boulder 2/03 F. Chong -- QC

Bit Flip Flips probabilities for |0> and |1> X Gate 1 a = b + a 1 X Bit-flip, Not 1 b Flips probabilities for |0> and |1> Conservation of energy Reversibility => unitary matrix (* means complex conjugate) Boulder 2/03 F. Chong -- QC

Controlled Not Control bit determines whether X operates 1 a Controlled Not a 00 + b 01 + 1 b Controlled X = 1 c d 10 + c 11 CNot X 1 d Control bit determines whether X operates Control bit is affected by operation Boulder 2/03 F. Chong -- QC

Universal Quantum Operations H Gate H Hadamard T Gate T Z Gate Z Phase-flip Controlled Not Controlled X CNot X Boulder 2/03 F. Chong -- QC

Quantum Algorithms Search (function evaluation) Factorization (FFT, discrete log) Adiabatic optimization (quantum simulated annealing) Estimating Gauss sums, Pell’s equation Testing matrix multiply Key distribution Digital signatures Clock synchronization Boulder 2/03 F. Chong -- QC

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 > 0 such that xr mod N = 1, x coprime N Then (xr/2 +/- 1) mod N = p,q eg Order(2, 15) = 4 (x4/2 +/- 1) mod 15 = 3,5 Boulder 2/03 F. Chong -- QC

Shor’s Algorithm [Shor94] m qubits H r r 0 xr mod N FT s/r |0> n qubits j |0> j=1: |r> = |0> + |4> + |8> + |12> + … j=2: |r> = |1> + |5> + |9> + |13> + … j=4: |r> = |2> + |6> + |10> + |14> + … j=8: |r> = |3> + |7> + |11> + |15> + … Boulder 2/03 F. Chong -- QC

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 Reduce fraction I*s/r => r is the denominator with high probability! Repeat algorithm if pq not equal N O(n3) instead of O(2n) !!! Boulder 2/03 F. Chong -- QC

Error Correction is Crucial Need continuous error correction can operate on encoded data [Shor96, Steane96, Gottesman99] Threshold Theorem [Ahanorov 97] failure rate of 10-4 per op can be tolerated Practical error rates are 10-6 to 10-9 Boulder 2/03 F. Chong -- QC

Quantum Error Correction X12 X23 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) Boulder 2/03 F. Chong -- QC

Syndrome Measurment X X X Y Y Y Y 12 ' 2 2 ' 1 1 Boulder 2/03 X 12 Y Y ' 2 2 Y Y ' 1 1 Boulder 2/03 F. Chong -- QC

3-bit Error Correction X X A X X X A X Y X Y Y X Y Y X Y Boulder 2/03 1 01 X X A X 12 Y X Y ' 2 2 Y X Y ' 1 1 Y X Y ' Boulder 2/03 F. Chong -- QC

Concatenated Codes Boulder 2/03 F. Chong -- QC

Error Correction Overhead 7-qubit code [Steane96], applied recursively Recursion Storage Operations Min. time (k) (7k) ( 153k ) ( 5k ) 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 Boulder 2/03 F. Chong -- QC

Recursion Requirements Shor’s Grover’s Boulder 2/03 F. Chong -- QC

Clustering Recursive scheme is overkill Don’t error correct every operation [Oskin,Chong,Chuang IEEE Computer 02] Boulder 2/03 F. Chong -- QC

Memory Hierarchy [Copsey,Oskin,Chong,Chuang,Abdel-Gaffar NSC02] Boulder 2/03 F. Chong -- QC

Fundamental Constraint: Quantum gates require classical control lines! ( Yablonovitch, 1999 ) ( Nakamura, Nature 398, p. 786, ‘99 ) ( Marcus 1997 ) Quantum: 20 nm Classical: 100’s of nm Boulder 2/03 F. Chong -- QC

Narrow control wires don’t work 5nm access points contain only a handful of quantum states at temp < 1K 20 nm Boulder 2/03 F. Chong -- QC

Perhaps a solution? As two physical dimensions of the access point exceed 100nm thousands of electron states are held. Classically, these states are restricted to the access point, however, quantum mechanically they tunnel downward, guided by the via, thus enabling control. Boulder 2/03 F. Chong -- QC

Classical meets Quantum Pitch-matching problem! 5nm Classical access points 100nm 100nm 100nm Natural architecture: linear arrays of qubits Narrow-tipped control 100nm Phosphorus Atoms 20nm 20nm Boulder 2/03 F. Chong -- QC

What about wires? A short quantum wire Short wire constructed from swap gates Each step requires 3 CNOT ops (swap) Key difference from classical: qubits are stationary Boulder 2/03 F. Chong -- QC

Why short wires are short How long before error correction? Limited by decoherence Threshold theorem => distance Theoretically  70 um (really 100X less) Very coarse bounds so far Boulder 2/03 F. Chong -- QC

How to get longer wires?? Use “Quantum Teleportation” Sender |a> Receiver 1 qubit 2 classical bits Boulder 2/03 F. Chong -- QC

CAT State Two bits are in “lockstep” Named for Shrodinger’s cat both 0 or both 1 Named for Shrodinger’s cat Also “EPR pair” for Einstein, Podolsky, Rosen Boulder 2/03 F. Chong -- QC

Teleportation Circuit source H |a> |b> EPR Pair (CAT) CNOT target X Z |c> |a> Source generates |bc> EPR pair Pre-communicate |c> to target with retry Classical communication to set value Can be used to convert between codes! Boulder 2/03 F. Chong -- QC