QUANTUM COMPUTING Part II Jean V. Bellissard Georgia Institute of Technology & Institut Universitaire de France

Hello again everyone !

QUANTUM GATES: a reminder

Quantum gates: U |x> U|x> 0 1 X = 1 0 0 -i Y = i 0 Z = 0 -1 I = 1-qubit gates U |x> U|x> U is unitary in M2 ( C ) 0 1 X = 1 0 0 -i Y = i 0 Z = 0 -1 I = 0 1 Pauli basis in M2 ( C )

Quantum gates: U |x> U|x> 1 1 H =2-1/2 1 -1 1 0 S = 0 i 1 0 T = 1-qubit gates U |x> U|x> U is unitary in M2 ( C ) 1 1 H =2-1/2 1 -1 1 0 S = 0 i 1 0 T = 0 eip/4 Hadamard, phase and p/8 gates

Quantum gates: |x> |x> U |y> Ux|y> controlled gates U is unitary in M2 ( C )

Quantum gates: the CNOT gate |x> |x> |y> |xÅy>

Quantum gates: the swap gate |x> x |y> = |y> x |x>

quantum computers are fast ! FOURIER TRANSFORM: quantum computers are fast !

Fourier Transform: Digital basis given by qubits |x1x2…xn> = |x1> |x2> … |xn> =|y> If y = 2(n-1) x1 + 2(n-2)x2 +…+ xn:= x1x2…xn

F |j> = 1 ∑k=0 e2ip jk/N |k> Fourier Transform : Fourier transform: F |j> = 1 ∑k=0 e2ip jk/N |k> N1/2 N=2n,

Fourier Transform : Binary decomposition: (modulo 1) where jk/2n = (0.jn)k1+ (0.jn-1jn)k2 + … + (0.j1j2…jn)kn (modulo 1) where 0.j1j2…jr = j1/2 + j2/22 …+ jr/2r

F |j> = 1 ∑k=0 e2ip jk/ |k> Fourier Transform : Binary decomposition: F |j> = 1 ∑k=0 e2ip jk/ |k> 2n/2 (|0> + e2ip(0.jn) |1>) (|0> + e2ip(0.j1…jn) |1>) . . . F|j> = 2n/2

Fourier Transform : Digital phase gate (1 qubit): 1 0 Rk = 0 e 1 0 Rk = 0 e 2ip 2k Rk|x> = e |x> 2ip x 2k

Fourier Transform : |jn> |jn-1> |j2> |j1> (|0> + e2ip(0.jn) |1>) H Rn Rn-1 R2 |jn-1> (|0>+e2ip(0.jn-1 jn) |1>) H Rn-2 |j2> (|0>+e2ip(0.j2...jn)|1>) H R2 |j1> (|0>+e2ip(0.j1...jn) |1>) H Circuit producing the quantum Fourier transform

Fourier Transform : Swap gates arrange final qubits in right order x (|0> + e2ip(0.jn) |1>) x (|0>+e2ip(0.j1...jn) |1>) (|0>+e2ip(0.jn-1 jn) |1>) x (|0>+e2ip(0.j2...jn)|1>) (|0>+e2ip(0.j2...jn)|1>) x (|0>+e2ip(0.jn-1 jn) |1>) (|0>+e2ip(0.j1...jn) |1>) x (|0> + e2ip(0.jn) |1>) Swap gates arrange final qubits in right order

Fourier Transform : ~ ~ Fourier transform F ∑j f(j)|j> = ∑kf(k)|k> f(k) = 2-n/2∑jf(j) e2ipjk/ the Fourier transform of f is given by the coordinates of the outcome. It can then be measured ~ ~ 2n

Fourier Transform : The usual FFT requires a time O(N LnN) The number of gates needed is n2/2 + 2n Since the N=2n, the algorithm gives the result in a time (1 time unit/gate) O((LnN)2) !!

PHASE ESTIMATION a key subroutine

Phase estimation U is a unitary with an eigenvalue U|u> = eif |u> Goal: compute f . Set-up: two registers, one with t-qubits, the other one for representing U.

GUn|x> |u> = einxf |x> |u> Phase estimation a controlled Un-gate GUn gives GUn|x> |u> = einxf |x> |u> It transfers the phase of |u> on the component |1> of the first register. On the first register one uses a rotated state H|0> = (|0>+|1>)/√2 instead of |x>.

Phase estimation |0> |0> |0> |0> |u> |u> |0> + ei2 f|1> (t-1) |0> H U 2(t-1) |0> + ei2 f|1> 2 |0> H U 22 |0> + ei2 f|1> 1 |0> H U 21 |0> + ei2 f|1> |0> H U 20 |u> |u>

Phase estimation . . . If f= 2p .j1j2…jt…, the outcome is Then use a Fourier transform back to get |j> =| j1j2…jt >, giving the value of the phase modulo O(2p/2t). If f= 2p .j1j2…jt…, the outcome is (|0> + e2ip(0.jt) |1>) (|0> + e2ip(0.j1…jt) |1>) . . . 2n/2

Phase estimation To get n digit of f accurate, with probability of success (1-e), it can be shown that t must be chosen as t=n+log(2+1/2e)

factorizing integer into primes SHOR’S ALGORITHM: factorizing integer into primes

Shor’s algorithm Input: a composite integer N Output: a non trivial factor of N Runtime: O((log N)3) operations, succeeds with probability O(1).

Shor’s algorithm First step: order finding. If x<N are integers with no common factors, the order of x modulo N is the least 0<r such that xrº1(mod N). Use the unitary U|y> = |xy(mod N)>. If y Î {0,1}L, N<2L, and N≤y<2L, set U|y> =|y>.

Shor’s algorithm Then |us> = r-1/2åk=0r exp(-2ipsk/r)|xr(mod N)> is an eigenvector of U with phase f=2p s/r A phase-finding computes s/r. A continuous fraction expansion gives r.

Shor’s algorithm It may not be possible to prepare the initial state of the second register in the state |us>. But any initial state is a linear combination of the |us> ‘s. The outcome will be s/r for some s. A continuous fraction expansion will give r anyway.

Shor’s algorithm Factoring procedure (i) If N is even, return the factor m=2 (ii) Find if N=ab, for a>1, b≥2, integers (special subroutine) (iii) Choose randomly xÎ[1,N-1]. If m=gcd(x,N) >1, then return m.

Shor’s algorithm Factoring procedure (continued): (iv) Find the order r of x mod N. (v) If r is even & xr/2-1≠-1 (mod N), compute gcd(xr/2-1,N) & gcd(xr/2=1,N), check if one is a nontrivial factor m. If so return m.

can quantum information be protected ? ERROR-CORRECTIONS: can quantum information be protected ?

Error-correction codes Classical code theory uses redundancy to transmit bits of information 1 000 111 010 110 000 111 Transmission Reconstruction at reception (correction) coding errors (2nd Law)

Error-correction codes Quantum computer are submitted to the no-cloning theorem! there is no Hilbert space H neither any unitary operator U on H H for which there is a state |s> such that U|y> |s> = |y> |y> ™ yÎ H

Error-correction codes However it is possible to produce quantum circuits for which |0>®|000> and |1>®|111> for instance: a|0>+b|1> |0> a|000>+b|111>

Error-correction codes The previous circuit protects against index flips. How can one protects the signal against phase flips ? Hadamard gates transform index into a phase: H|x> = (|0>+(-1)x|1>)/√2

Error-correction codes Phase flip protection a|0>+b|1> |0> H ∑y(a+b(-1)∑yi)|y1y2y3> 2-3/2 a|000>+b|111>

Error-correction codes a|0>+b|1> |0> H Shor’s code

Error-correction codes Shor’s code gives |0>®|0L> and |1>®|1L> with: |xL>=________________________ 2√2 (|000>+(-)x|111>)(|000>+(-)x|111>)(|000>+(-)x|111>)

Error-correction codes Kitaev proposed in 1997 to replace digital degrees of freedom by topological ones. Tunneling effect between topological sectors is unlikely, leading to a better code protection.

PHYSICAL REALIZATIONS can quantum computers be built ?

Realizations Several devices may produce qubits: Any quantum harmonic oscillator Optical photons Optical cavity quantum electrodynamics: coupling with 2-level atoms. Ion traps Nuclear magnetic resonance: computation with up to 7-qubits have permitted to test Shor’s algorithm 15=3x5 !! Josephson junctions: quantronium Double well with quantum dots

Realizations: 1-qubit, the quantronium The quantronium (Esteve & Devoret Saclay): a Josephson tunneling junction

Realizations 1-qubit, the quantronium

Realizations Quantronium : RABI OSCILLATIONS Coherent manipulation of the Quantronium state: a microwave resonant pulse with duration t and amplitude URF is applied to the gate. The Quantronium undergoes Rabi oscillations. The probability of measuring the Quantronium in its excited state, i.e. the switching probability of the measuring junction, oscillates accordingly as a function of t and URF. Each dot is an average over 50000 measurements. The decoherence time is about 5µs.

Realizations 1-qubit, quantum dots Double quantum dots : group of Kouwenhoven, (U. Delft Holland) resonant tunneling µR µL -|e|V -|e|f1(N1+1,N2) -|e|f2(N1, N2+1) 1 2 3 4 5 a b

Realizations 7-qubit, NMR Nuclear Magnetic Resonance : IBM 15=3x5 !! (Shor’s algorithm)

will quantum computers be built ? CONCLUSIONS will quantum computers be built ?

To conclude (from Part I) The elementary unit of quantum information is the qubit, with states represented by the Bloch ball. Several qubits are given by tensor products leading to entanglement. Quantum gates are given by unitary operators and lead to quantum circuits Law of physics must be considered for a quantum computer to work: measurement, dissipation…

To conclude (Part II) Several algorithms are available: Fourier transform, phase estimation, quantum search, hidden subgroup, order-finding Shor’s algorithm for factoring shows enormous efficient and threaten present cryptography Error-correcting codes are now available Few qubits computer have been realized with NRM experiments

To conclude (other topics) A theory of quantum information and code theory is also available even though incomplete Quantum cryprography exists (Gisin, Geneva) Need for developments in quantum complexity theory: are notions of P- NP- completeness obsolete ? Main problem: putting qubits together in concrete machines. Can one control entanglement and /or decoherence on a large scale ? … Not clear !!

Will quantum computers be built ? YES of course !!