Quantum computing and qubit decoherence Semion Saikin NSF
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. 2
Quantum computation Modeling of quantum systems 1 particle – n equations: R. Feynman, Inter. Jour. Theor. Phys. 21, 467 (1982) L particles – n. L equations! 3
Quantum computation Applications • Modeling of quantum systems Pharmaceutical industry Nanoelectronics • Quantum search algorithm L. Grover (1995) • Factorization of large integer numbers P. Shor (1994) RSA Code: Military, Banking • Quantum Cryptography Process optimization: Industry Military Bob Alice Eve
Quantum computation Bit & Qubit • Two states classical bit 1 • Two levels quantum system (qubit) 0 Polarization vector: S=(Sφ Sθ SR=const) Density matrix: • Equalities 1 • Single qubit operations ≡ 0 ≡ 5
Quantum computation Entanglement + = + ≠ Non-separable quantum states: 6
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. I n p u t Unitary transformation Classical control O u t p u t D. P. Di. Vincenzo, G. Burkard, D. Loss, E. V. Sukhorukov, cond-mat/9911245 7
QC Roadmap http: //qist. lanl. gov/ Quantum computation Physical realization of a qubit • Ion traps and neutral atoms • Semiconductor charge qubit Single QD Double QD E 2 e E 1 E 0 • Photon based QC E 1 E 0 P • Spin qubit • Superconducting qubit Cooper pair box e SQUID Nuclear spin (liquid state NMR, solid state NMR) I Electron spin S i N pairs - N+1 pairs - 8
Quantum computation Decoherence. Interaction with macroscopic environment. Markov process T 1 T 2 concept Non-exponential decay 0 t t
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 10
Donor electron spin in Si: P Structure Si atom (group-IV) Diamond crystal structure Natural Silicon: 28 Si – 92% – 4. 7% 30 Si – 3. 1% 29 Si 5. 43Å I=1/2 31 P electron spin (T=4. 2 K) T 1~ min T 2~ msecs P atom (group-V) + = b ≈ 15 Å a ≈ 25 Å Natural Phosphorus: 31 P – 100% I=1/2 In the effective mass approximation electron wave function is s-like: 11
Donor electron spin in Si: P Application for QC A - gate Bohr Radius: Si: a ≈ 25 Å b ≈ 15 Å J - gate Ge: a ≈ 64 Å b ≈ 24 Å Six. Ge 1 -x Si Si 1 -x. Gex B. E. Kane, Nature 393 133 (1998) 31 P donor Qubit – nuclear spin Qubit-qubit inteaction – electron spin S 1 HEx J - gate S 2 R. Vrijen, E. Yablonovitch, K. Wang, H. W. Jiang, A. Balandin, V. Roychowdhury, T. Mor, D. Di. Vincenzo, Phys. Rev. A 62, 012306 (2000) 31 P donor Qubit – electron spin Qubit-qubit inteaction – electron spin HHf A - gate S 1 I 2 I 1 Qubit 2 Qubit 1 HEx S 2 Qubit 2 12
Donor electron spin in Si: P Sources of decoherence • Interaction with phonons D. Mozyrsky, Sh. Kogan, V. N. Gorshkov, G. P. Berman Phys. Rev. B 65, 245213 (2002) • Gate errors X. Hu, S. Das Sarma, cond-mat/0207457 • Interaction with 29 Si nuclear spins Theory 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, R 1809 (2003) Experiments 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) 13
Donor electron spin in Si: P Spin Hamiltonian 28 Si H 31 P e 29 Si Effect of external field Electronnuclei interaction Nucleinuclei interaction Electron spin Zeeman term: Effective Bohr radius ~ 20 -25 Å Lattice constant = 5. 43 Å Nuclear spin Zeeman term: In a natural Si crystal the donor electron interacts with ~ 80 nuclei of 29 Si System of 29 Si nuclear spins can be considered as a spin bath Hyperfine electron-nuclear spin interaction: Dipole-dipole nuclear spin interaction: 14
Donor electron spin in Si: P Hyperfine interaction Contact interaction: e- Dipole-dipole interaction: 29 Si Hyperfine interaction: Approximations: Contact interaction only: High magnetic field Contact interaction High magnetic field 15
Donor electron spin in Si: P Energy level structure (high magnetic field) H - 31 P electron spin - 31 P nuclear spin - 29 Si nuclear spin … 16
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) 17
Donor electron spin in Si: P Effects of nuclear spin bath (high field) (a) S=“ ” (b) S=“ ” e- “ - pulse” + e- Electron spin system Hz Hz Nuclear spin system Heff Ik 28 Si H 31 P Hz Ik 29 Si H 31 P 18
Donor electron spin in Si: P Hyperfine modulations of an electron spin qubit || || t Threshold value of the magnetic field for a fault tolerant 31 P electron spin qubit: S. Saikin and L. Fedichkin, Phys. Rev. B 67, article 161302(R), 1 -4 (2003) 19
Donor electron spin in Si: P Spin echo modulations. Experiment. Spin echo: A( ) Hx Mx M. Fanciulli, P. Hofer, A. Ponti Physica B 340– 342, 895 (2003) Si-nat 0 2 t E. Abe, K. M. Itoh, J. Isoya S. Yamasaki, cond-mat/0402152 T = 10 K H || [0 0 1] 20
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. 21
Prospects for future • Spin diffusion A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring Phys. Rev. B 68, 193207 (2003) • Control for spin-spin coupling in solids • Initial drop of spin coherence M. Fanciulli, P. Hofer, A. Ponti Physica B 340– 342, 895 (2003) Developing of error avoiding methods for spin qubits in solids. S. Barrett’s Group, Yale M. Fanciulli’s Group, MDM Laboratory, Italy 22
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 Task: Identify measures of decoherence and establish their approximate “additivity” for several qubits Task: Apply to 2 DEG and other QC designs; improve or discard QC designs and measurement schemes Use perturbative Markovian schemes Relaxation time scales: T 1, T 2, and additivity of rates QHE QC P in Si QC Q-dot QC New short-time approximations (De)coherence in Transport “Deviation” measures of decoherence and their additivity QHE QC P in Si QC Q-dot QC Measurement by charge carriers Coherent spin transport How to measure spin and charge qubits Spin polarization relaxation in devices / spintronics Improve and finalize solid-state QC designs once the single-qubit measurement methodology is established 23
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