Quantum computing and qubit decoherence Semion Saikin NSF Center for Quantum Device Technology Clarkson University

Outline Quantum computation. Modeling of quantum systems Applications Bit & Qubit Entanglement Stability criteria Physical realization of a qubit Decoherence Measure of Decoherence Donor electron spin qubit in Si:P. Effect of nuclear spin bath. Structure Application for Quantum computation Sources of decoherence Spin Hamiltonian Hyperfine interaction Energy level structure (high magnetic field) Effects of nuclear spin bath (low field) Effects of nuclear spin bath (high field) Hyperfine modulations of an electron spin qubit Conclusions. Prospects for future.

Quantum computation Modeling of quantum systems 1 particle – n equations: L particles – nL equations! R. Feynman, Inter. Jour. Theor. Phys. 21, 467 (1982)

Quantum computation Applications Modeling of quantum systems Factorization of large integer numbers P. Shor (1994) Pharmaceutical industry Nanoelectronics RSA Code: Military, Banking Quantum search algorithm L. Grover (1995) Quantum Cryptography Alice Bob Eve Process optimization: Industry Military

1 1 ≡ ≡ Quantum computation Bit & Qubit Two states classical bit Two levels quantum system (qubit) Polarization vector: 1 S=(Sφ Sθ SR=const) Density matrix: Equalities Single qubit operations 1 ≡ ≡

Quantum computation Entanglement + = ≠ + ≠ + Non-separable quantum states:

Quantum computation Stability criteria The machine should have a collection of bits. (~103 qubits) It should be possible to set all the memory bits to 0 before the start of each computation. The error rate should be sufficiently low. (less 10-4 ) It must be possible to perform elementary logic operations between pairs of bits. Reliable output of the final result should be possible. O u t p I n p u t Unitary transformation D. P. DiVincenzo, G. Burkard, D. Loss, E. V. Sukhorukov, cond-mat/9911245 Classical control

Physical realization of a qubit Quantum computation QC Roadmap http://qist.lanl.gov/ Physical realization of a qubit Ion traps and neutral atoms Semiconductor charge qubit Single QD Double QD E0 E1 E2 e e Photon based QC E1 E0 P Spin qubit Superconducting qubit Cooper pair box Nuclear spin (liquid state NMR, solid state NMR) Electron spin SQUID  S i I N pairs - N+1 pairs -

Decoherence. Interaction with macroscopic environment. Quantum computation Decoherence. Interaction with macroscopic environment. Markov process T1 T2 concept Non-exponential decay t t

Measure of Decoherence Quantum computation Measure of Decoherence Basis independent. Additive for a few qubits. Applicable for any timescale and complicated system dynamics. S ideal S real A. Fedorov, L. Fedichkin, V. Privman, cond-mat/0401248

Donor electron spin in Si:P Structure Si atom (group-IV) Diamond crystal structure Natural Silicon: 28Si – 92% 29Si – 4.7% I=1/2 30Si – 3.1% 5.43Å 31P electron spin (T=4.2K) T1~ min T2~ msecs P atom (group-V) = b ≈ 15 Å + a ≈ 25 Å Natural Phosphorus: 31P – 100% I=1/2 In the effective mass approximation electron wave function is s-like:

Donor electron spin in Si:P Application for QC Bohr Radius: Si: a ≈ 25 Å b ≈ 15 Å Ge: a ≈ 64 Å b ≈ 24 Å Si1-xGex SixGe1-x A - gate J - gate Si B.E.Kane, Nature 393 133 (1998) R.Vrijen, E.Yablonovitch, K.Wang, H.W.Jiang, A.Balandin, V.Roychowdhury, T.Mor, D.DiVincenzo, Phys. Rev. A 62, 012306 (2000) 31P donor Qubit – nuclear spin Qubit-qubit inteaction – electron spin 31P donor Qubit – electron spin Qubit-qubit inteaction – electron spin HEx J - gate S1 S2 HHf A - gate S1 HEx S2 I1 I2 Qubit 1 Qubit 2 Qubit 1 Qubit 2

Donor electron spin in Si:P Sources of decoherence Interaction with phonons Gate errors Interaction with 29Si nuclear spins Theory Experiments D. Mozyrsky, Sh. Kogan, V. N. Gorshkov, G. P. Berman Phys. Rev. B 65, 245213 (2002) X.Hu, S.Das Sarma, cond-mat/0207457 I.A.Merkulov, Al.L.Efros, M.Rosen, Phys. Rev. B 65, 205309 (2002) S.Saikin, D.Mozyrsky, V.Privman, Nano Letters 2, 651 (2002) R. De Sousa, S.Das Sarma, Phys. Rev. B 68, 115322 (2003) S.Saikin, L. Fedichkin, Phys. Rev. B 67, 161302(R) (2003) J.Schliemann, A.Khaetskii, D.Loss, J. Phys., Condens. Matter 15, R1809 (2003) A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring, Phys. Rev. B 68, 193207 (2003) M. Fanciulli, P. Hofer, A. Ponti, Physica B 340–342, 895 (2003) E. Abe, K. M. Itoh, J. Isoya S. Yamasaki, cond-mat/0402152 (2004)

Donor electron spin in Si:P Spin Hamiltonian 28Si H e- 31P Effect of external field Electron- nuclei interaction Nuclei- nuclei interaction 29Si Electron spin Zeeman term: Effective Bohr radius ~ 20-25 Å Lattice constant = 5.43 Å In a natural Si crystal the donor electron interacts with ~ 80 nuclei of 29Si System of 29Si nuclear spins can be considered as a spin bath Nuclear spin Zeeman term: Hyperfine electron-nuclear spin interaction: Dipole-dipole nuclear spin interaction:

Donor electron spin in Si:P Hyperfine interaction Contact interaction: Dipole-dipole interaction: e- 29Si Hyperfine interaction: Approximations: Contact interaction High magnetic field Contact interaction only: High magnetic field

Donor electron spin in Si:P Energy level structure (high magnetic field) H - 31P electron spin - 31P nuclear spin - 29Si nuclear spin …

Donor electron spin in Si:P Effects of nuclear spin bath (low field) S. Saikin, D. Mozyrsiky and V. Privman, Nano Lett. 2, 651-655 (2002)

Donor electron spin in Si:P Effects of nuclear spin bath (high field) (a) S=“” (b) S=“” e- “ - pulse” e- + Hz Electron spin system 31P Hz Heff 29Si Ik H Nuclear spin system 31P 28Si Hz H Heff 29Si Ik

Donor electron spin in Si:P Hyperfine modulations of an electron spin qubit |||| t Threshold value of the magnetic field for a fault tolerant 31P electron spin qubit: S. Saikin and L. Fedichkin, Phys. Rev. B 67, article 161302(R), 1-4 (2003)

Donor electron spin in Si:P Spin echo modulations. Experiment. Spin echo: t Hx Mx  2 A() M. Fanciulli, P. Hofer, A. Ponti Physica B 340–342, 895 (2003) Si-nat T = 10 K H || [0 0 1] E. Abe, K. M. Itoh, J. Isoya S. Yamasaki, cond-mat/0402152

Conclusions Effects of nuclear spin bath on decoherence of an electron spin qubit in a Si:P system has been studied. A new measure of decoherence processes has been applied. At low field regime coherence of a qubit exponentially decay with a characteristic time T ~ 0.1 sec. At high magnetic field regime quantum operations with a qubit produce deviations of a qubit state from ideal one. The characteristic time of these processes is T ~ 0.1 sec. The threshold value of an external magnetic field required for fault-tolerant quantum computation is Hext ~ 9 Tesla.

Developing of error avoiding methods for spin qubits in solids. Prospects for future Spin diffusion Initial drop of spin coherence M. Fanciulli, P. Hofer, A. Ponti Physica B 340–342, 895 (2003) A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring Phys. Rev. B 68, 193207 (2003) Developing of error avoiding methods for spin qubits in solids. Control for spin-spin coupling in solids S. Barrett’s Group, Yale M. Fanciulli’s Group, MDM Laboratory, Italy

NSF Center for Quantum Device Technology PI V. Privman Modeling of Quantum Coherence for Evaluation of QC Designs and Measurement Schemes Task: Model the environmental effects and approximate the density matrix Use perturbative Markovian schemes New short-time approximations (De)coherence in Transport “Deviation” measures of decoherence and their additivity Measurement by charge carriers Measurement by charge carriers Coherent spin transport Coherent spin transport Task: Identify measures of decoherence and establish their approximate “additivity” for several qubits Relaxation time scales: T1, T2, and additivity of rates How to measure spin and charge qubits Spin polarization relaxation in devices / spintronics Task: Apply to 2DEG and other QC designs; improve or discard QC designs and measurement schemes QHE QC P in Si QC Q-dot QC QHE QC P in Si QC Q-dot QC Improve and finalize solid-state QC designs once the single-qubit measurement methodology is established