1. A Study of Error-Correcting Codes for Quantum Adiabatic Computing Omid Etesami Daniel Preda CS252 – Spring 2007

2. Quantum Computing Some highlights: Feynman: Polynomial-time simulation of quantum mechanical systems Shor: polynomial-time integer factoring algorithm Grover: Squared speed-up of unstructured search Brassard-Bennett: Quantum public key distribution

3. Quantum Circuit Model System of n qubits described by a unit-length 2^n- dimensional complex vector. Evolution of the system: U1U1U1U1 …. U5U5U5U5 U4U4U4U4 U3U3U3U3 U2U2U2U2 Outputmeasure Gate = Unitary operator

4. Schrodinger equation Hamiltonian is a Hermitian matrix

5. Adiabatic Quantum Computing: alternative architecture HiHi Hamiltonian with easily-prepared ground state HfHf Ground state encodes solution to Max-2SAT instance (1-s)H i +sH f Quantum analogue of simulating annealing Hamiltonians sum of local interactions Running time depends on the energy gap between lowest and second lowest eigenvalue (~ 1/gap^2)

6. Theoretical Universality H i+1j H ij H ij+1 H’ ij H HH H H H H H  L gates time T=poly(L) Quantum circuit model = adiabatic computation with local interactions on 2-D lattice

7. Errors in Adiabatic Computing? For quantum circuit model, quantum error- correcting codes have been designed. Computation is perfect below a certain error rate threshold. For adiabatic computation, the system evolves continuously over a long time. Errors can propagate!

8. Error-Correction for Adiabatic: Jordan, Farhi, Shor [06] Stabilizer code Encoding of qubits: (n qubits to 4n qubits) Encoding of Pauli operators of original Hamiltonian:

9. JFS [06] (continued) Add penalty Hamiltonian Intuition: this extra Hamiltonian penalizes all states that are not within the code-space Resilient against 1-local errors (can be extended to 2- local):

10. Implementation Prototype Problem: MAX-2SAT (x or y) & (x or !y) & (!y or z) & (!x or z) & (!y or !z) It is NP-Complete Reduce problem to final Hamiltonian H: ground energy of H = min # violated clauses Ground state of initial Hamiltonian = random superposition of all assignments Generate random instances We have to map the initial state, and the initial and final Hamiltonians into the larger space, and then we use the inverse map to read the final measured state.

11. Implementation Solve Shrodinger equation using Runge-Kutta method of order 6 Numerical techniques for controlling simulation errors Change Hamiltonian uniformly in time

12. Parameters and Error model Error model: bit flip on the system; no decoherence by the environment T: running time of evolution e: strength of error Hamiltonian, which is the rate of bit flip. Represents the temparature. E_p: strength of error-correcting penalty Evaluation based on P: probabibility that the measured state is optimal Final number of qubits = 3 * 4 = 12

13. Also when e=0.5, and T is large, correctness probability converges to 1/8

15. Conclusions For small errors, no correction leads to good but not perfect results, whereas error-correction bounds final error to almost zero (less than 0.5%). For larger errors, no correction gives random (or even worse) solutions. By keeping the penalty/noise ratio large (like E_p/e around 10), the error-correction still gives almost good solutions (error rate less than 1%). Looks like larger errors require smaller E_p/e ratio.