Autonomous Quantum Error Correction Joachim Cohen QUANTIC
Outline I. Why quantum information processing? II. Classical bits vs. Quantum bits III. Quantum Error Correction
WHY QUANTUM INFORMATION PROCESSING?
Why quantum information processing? Cryptography : quantum key distribution Quantum simulator : simulation of quantum systems Quantum algorithms : Shor s algorithm on prime number factorization in polynomial time
CLASSICAL BITS VS. QUANTUM BITS
Classical PHYSICAL Bit BIT = : BISTABLE Bistable SYSTEM system Mechanical system with electrical readout: switch! Bit : State 0 ou 1 1 Electrical system with electrical readout: RAM cell +V cc 0 1 1 0 0 Intérupteur électrique CMOS Transistors: + + N N P P Courtesy of Michel Devoret, Collège de France, 2010 Curtesy of Michel Devoret, Collège de France, 2010 Cellule de RAM 10-I-8
Classical Bit : Bistable system U(x) 0 1 ΔU noise k B T x m d2 x dt 2 + dx dt + @U @x =0 : fric8on coefficient friction thermal noise k B T<<ΔU
Quantum physics? Quantum bit (Qubit)?
Spring : Classical case k x m d 2 x dt 2 +! 2 x =0 x(t) =Acos(!t + ) Harmonic potential V (x) = 1 2 kx2 No dissipation : constant energy E Small A 2 Large A 2 x
Quantum Mechanics : System energy is quantified! Example : Light photon photon photon 1, 2, 3,, N,! Energy = N*hω! Light energy is quantified! photodetector
Spring : Quantum case Discrete set of stationnary energy states : ψ 0 (x), ψ 1 (x), ψ 2 (x), ψ 3 (x)... L 2 functions associated to energies E 0, E 1, E 2, E 3... k satisfying [ ~ 2 2m d 2 dx 2 + U(x)] k = E k k lψ(x)l 2 = density of probability to find the system in x Restrict to the first two energy levels : ħω E 2 3 and 0i := 0 1i := 1 ψ 1 (x) ħω/2 0 1 x ψ 0 (x) x
Postulate 1 : Quantum superposition { 0i, 1i} form an orthonormal basis of a 2D-Hilbert space with h0 1i = R D dx 0(x) 1 (x) Quantum superposition : general state is given by i = 0i + 1i 2 + 2 =1 - - 2 2, 2 C probability to find the system in state probability to find the system in state 0i 1i Qubit! 0i 1i 0i 1i p 2 0i+2i 1i p 5 0i 1i
Postulate 2 : Quantum Measurement Quantum measurement : Consider the qubit in state i = 0i + 1i Ask the system : are you in 0ior 1i? With probability 2 the answer is 0i The system is projected in state i = 0i! Measurement modifies the qubit state! Quantum measurement can be DESTRUCTIVE!
Composite system and Quantum Entanglement Composite system : Consider two qubits A and B Qubit A lives in H A Qubit B lives in H B Joint system qubits A+B lives in H A H B H A H B =vec C { 00i, 10i, 01i, 11i} 00i := 0i A 0i B 01i := 0i A 1i B A B Entangled state : i = p 1 2 00i + p 1 2 11i 2 H A H B 0i Consider First, we measure qubit A : we find qubit A in (with 50% probability ) The joint state collapses to 0i i = 00i qubit B is in with probability 1!
Quantum «rules» : Summary 1) Quantum superposition : general qubit state i = 0i + 1i 2 + 2 =1, 2 C 2) Measurement : revealing information about the state can destroy the superposition 3) Quantum Entanglement : possibility of having strongly correlated states between two qubits
Consequence : Decoherence Unwanted coupling with the environment : Qubit Environment -The environment measures the qubit and this measurement destroys the quantum superposition! -> lifetime of typically 100us (for superconducting circuits) - Lifetime decreases with the number of qubits How can we fight decoherence?
QUANTUM ERROR CORRECTION (QEC)
Bit vs. Qubit errors Errors on classical bits : bit-flip errors 0 1 1 0 Errors on qubits : i = 0i + 1i 0 i = 0 0i + 0 1i Errors can be cast in two error channels : Bit-flip errors 0i 1i 1i 0i Rest of the talk Phase-flip errors 0i 0i 1i - 1i p1 1 2 [ 0i + 1i] p 2 [ 0i 1i]
Quantum error correction Classical error correction: information redundantly encoded Ex : 0 000 1 111 such that error on bit 1 : 100 000 error on bit 2 : 101 111
Quantum error correction Quantum error correction (bit-flip errors only) three-qubit bit-flip code : 0i 1i 000i 111i 0i + 1i 000i + 111i Error on qubit 1 : 100i + 011i 000i + 111i But information about and must not be revealed... = How do we detect errors without destroying the state? = 0 L i 1 L i
Quantum error correction Error detection : Parity measurement qubit 1 flips 000i + 111i 100i + 011i P 12 = 0 P 23 = 0 Do not measure : Single parities P 1, P 2, P 3 What we can measure : Joint parities P 12 := [Q1+Q2] mod 2 P 23 := [Q2+Q3] mod 2 NON-destructive measurements! P 12 = 1 P 23 = 0 010i + 101i P 12 = 1 P 23 = 1 001i + 110i P 12 = 0 P 23 = 1
Quantum error correction Error Correction : simply apply inverse operation 100i + 011i 000i + 111i P 12 = 0 P 23 = 0 flip qubit 2 P 12 = 1 P 23 = 0 010i + 101i P 12 = 1 P 23 = 1 001i + 110i P 12 = 0 P 23 = 1
Quantum error correction : implementation 1 st option Build a feedback loop real-time data analysis takes time quantum systems are short-lived Superconducting circuits 100 us Courtesy of Quantum Electronics group, LPA, ENS (Paris)
Quantum error correction : feedback loop Error Correction? Use a flipper! qubits system Measurement output Error syndrome P12 = 0,1 P23 = 0,1 feedback
Quantum error correction : implementation 2 nd option : (what we have proposed) Autonomous QEC by coupling the qubits with another strongly dissipative quantum system : Qubit Dissipative system designed coupling
Autonomous quantum error correction Main idea : Coherent stabilization of the manifold {l000>,l111>} through dissipation coupled system does not dis8nguish 001i 110i 010i 101i 100i 011i 000i + 111i x
In practice Complete codes (correct for all types of errors) exist but have never been physically implemented Qubits of many kinds : trapped ions, superconducting qubits, NV centers... Quantum computer : 10 qubits max so far. Limited by decoherence! -> QEC remains a challenge to overcome!
Thanks! Questions?