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Optimal control of the quantum gate operations for quantum computing

Hsi-Sheng Goan 管 希 聖. Optimal control of the quantum gate operations for quantum computing. with Dung-Bang Tsai and Po-Wen Chen. Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan. Ref: Phys. Rev. A 79 , 060306

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Optimal control of the quantum gate operations for quantum computing

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  1. Hsi-Sheng Goan 管 希 聖 Optimal control of the quantum gate operations for quantum computing with Dung-Bang Tsai and Po-Wen Chen Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan Ref:Phys. Rev. A 79, 060306 (Rapid Communications) (2009).

  2. Quantum Computation and Quantum Information • The study of the information processing and computing tasks that can be accomplished usingquantum mechanical systems. • To exploit quantum effects, based on the principles of quantum mechanics to compute and process information in ways that are faster or more efficient than or even impossible on conventional computers or information processing devices.

  3. RSA cryptography • The difficulty of factorizing large numbers forms the basis of RSA encryption system: standard industrial strength encryption on the Internet • Example: 4633 = 41 x 113 • RSA systems offers each prizes to people who factor number like (US $200K for this one): • Example:factor a 300-digit number; Best algorithm: takes 1024 steps; • On computer at THz speed: 150,000 years

  4. 編碼保密傳輸 解碼金鑰 編碼金鑰 • 網路銀行(internet banking): N = p q • Public key: 公開的編碼金鑰(N,e) • Private key:不公開的解碼金鑰(N, p,q)

  5. Peter Shor Quantum algorithms and computational speed-ups • Algorithm:a detailed step-by-step method for solving a problem • Computer:a universal machine that can implement any algorithm • Quantum factoring algorithm : exponential speed-up (Shor’s Algorithm) Example: factor a 300-digit number • Quantum search of an unsorted database: quadratic speed-up (Grover’s Algorithm) • Example: name  phone number (easy) • phone number  name (hard) • Classical: O(n), Grover’s: • Simulation of quantum systems: up to exponential speed-up.

  6. Quantum bits • Classical bit: 0 or 1; voltage high or low • Quantum bit (QM two-state system): • Spin states; • Charge states; left or right • Flux states; L or R • Energy states, ground or excited states • Photon polarizations; H or V; L or R • Photon number (Fock) states; • More …

  7. Requirements for physical implementation of quantum computation • A scalable physical system with well characterized qubits • The ability to initialize the state of the qubits to a simple fiducial state, such as |000……〉. • Long relevant decoherence times, much longer than the gate operation time • A universal set of quantum gates • A qubit-specific measurement capability

  8. Liquid-state NMR NMR spin lattices Linear ion-trap spectroscopy Neutral-atom optical lattices Cavity QED + atoms Linear optics with single photons Nitrogen vacancies in diamond Electrons on liquid He Small Josephson junctions “charge” qubits “phase” qubits “flux” qubits Impurity spins in semiconductors Coupled quantum dots Qubits: spin,charge,excitons Exchange coupled, cavity coupled Physical systems actively consideredfor quantum computer implementation

  9. Electron spins in quantum dots • Top electrical gates define quantum dots in 2DEG. • Coulomb blockade confines excessive electron number at one per dot. • Spins of electrons are qubits. • Qubits can be addressed individually: • Back gates can move electrons into magnetized or high-g layer to produce locally different Zeeman splitting. • Or a current wire can produce magnetic field gradient. • Exchange coupling is controlled by electrically lowing the tunnel barrier between dots

  10. Silicon-based quantum bits • Donor nuclear spins[Kane, Nature (1998)] • Donor electron spins • Si-Ge hetero-structures [Vrijen et al., PRA (2000)] • Dipolar coupling [de Sousa et al., PRA (2004)] • Surface gate and global control [Hill et al., (2005)] • Donor electron-nuclear spin pairs • Digital Approach [Skinner et al., PRL (2003)] • Donor electron charges • P/P+ charge qubit [Hollenberg et al., (2004)] • Electron spins in silicon-based quantum dots[Friesen et al., PRB (2002)]

  11. Silicon-based electron-mediated nuclear spin quantum computer B. Kane, Nature (1998) • Exploiting the existing strength of Si technology • Qubits are nuclear spins of P donors in a regular array in pure silicon • Low temperature: • Effective Hamiltonian involves only spins • Long spin coherence and relaxation times • Magnetic field B to polarized electron spins • Control with surface gates and NMR pulses • Donor separation ~ 20nm • Gate width < 10nm

  12. Phosphorus Donor in Si P donor behaves effectively like a hydrogen-like atom embedded in Si P shallow donor energy levels in Si

  13. Silicon-based quantum computing Two interactions: hyperfine and exchange interactions . Determining the strength of these two interactionsas function ofdonor depth, donor separation and surface gate configuration and voltage. • L.M. Kettle, H.-S. Goan, S.C. Smith, C.J. Wellard, L.C.L. Hollenberg and C.I. Pakes, “A numerical study of hydrogenic effective mass theory for an impurity P donor in Si in the presence of an electric field and interfaces'', Physical Review B 68, 075317 (2003). • C.J. Wellard, L.C.L. Hollenberg, F. Parisoli, L.M. Kettle, H.-S. Goan, J.A.L. McIntosh and D.N. Jamieson, “Electron exchange coupling for single donor solid-state spin qubits”, Physical Review B 68, 195209 (2003). • L.M. Kettle, H.-S. Goan, S.C. Smith, L.C.L. Hollenberg and C.J. Wellard, ”Effect of J-gate potential and interfaces on donor exchange coupling in the Kane quantum computer architecture'', Journal of Physics: Condensed Matter 16, 1011 (2004). • C.J. Wellard, L.C.L. Hollenberg, L.M. Kettle and H.-S. Goan, “Voltage control of exchange coupling in phosphorus doped silicon”,Journal of Physics: Condensed Matter 16, 5697 (2004). • L. M. Kettle, H.-S. Goan, and S. C. Smith, “Molecular orbital calculations of two-electron states for P donor solid-state spin qubits”, Physical review B 73, 115205 (2006).

  14. Quantum gate operation, and quantum algorithm modelling CNOT • C. D. Hill and H.-S. Goan, “Fast non-adiabatic two-qubit gates for the Kane quantum computer”, Physical Review A 68, 012321 (2003). • C.D. Hill and H.-S. Goan, “Comment on Grover search with pairs of trapped ions“, Physical Review A 69, 056301 (2004). • C.D. Hill and H.-S. Goan, “Gates for the Kane quantum computer in the presence of dephasing”, Physical Review A 70, 022310 (2004). • C. D. Hill, L. C. L. Hollenberg, A. G. Fowler, C. J. Wellard, A. D. Greentree, and H.-S. Goan,“Global control and fast solid-state donor electron spin quantum computing”, Physical Review B 72, 045350 (2005). • C.D. Hill and H.-S. Goan, “Fast non-adiabatic gates and quantum algorithms on the Kane quantum computer in the presence of dephasing”,AIP Conference Proceedings Vol. 734, pp167-170 (2004).

  15. The CNOT gate • After using some single qubit identities to simplify, this circuit becomes: • Under typical expected conditions, numerical simulation shows that the CNOT gate has a systematic error of 4.0 x 10-5 and takes a total time of 16.0 s. • Similar circuits can be found for any two qubit gate, including swap and square root of swap gates.

  16. Silicon-based quantum bits • Donor nuclear spins[Kane, Nature (1998)] • Donor electron spins • Si-Ge hetero-structures [Vrijen et al., PRA (2000)] • Dipolar coupling [de Sousa et al., PRA (2004)] • Surface gate and global control [Hill et al., (2005)] • Donor electron-nuclear spin pairs • Digital Approach [Skinner et al., PRL (2003)] • Donor electron charges • P/P+ charge qubit [Hollenberg et al., (2004)] • Electron spins in silicon-based quantum dots[Friesen et al., PRB (2002)]

  17. Donor electron spin in Si-Ge structure R. Vrijen et al, Concept device: spin-resonance transistor, Phys. Rev. A 62, 012306 (2000)

  18. Silicon-based electron-mediated nuclear spin quantum computer B. Kane, Nature (1998) • Exploiting the existing strength of Si technology • Qubits are nuclear spins of P donors in a regular array in pure silicon • Low temperature: • Effective Hamiltonian involves only spins • Long spin coherence and relaxation times • Magnetic field B to polarized electron spins • Control with surface gates and NMR pulses • Donor separation ~ 20nm • Gate width < 10nm

  19. Single-qubit system Effective low-energy low-temperature Hamiltonian: Energy separation: gemBB

  20. Single-qubit system Qubit energy separation (if nuclear spins is initialized in spin-up state): Effective single-qubit Hamiltonian: gemBB Hamiltonian in a Bac field:

  21. Single-qubit control • Having control over hyperfine interaction by applying voltage to A gate would allow us to: • Change the resonant frequency of a particular qubit. • Perform X and Y rotations on a specific qubit using a resonant magnetic field • Perform a Z on a specific qubit (much faster than X and Y rotations) • These three operations allow us to do any single qubit rotation on the nuclear spins. B. Kane, Nature 393, 133 (1998)

  22. Single qubit rotations Laboratory frame Reference frame P(wac) wac w0

  23. Rx(q) rotation in the global control donor e-spin QC • Set a detuned target qubit toperform a 2protation, and then every other spectator qubit will undergo a rotation around x-axis with an angle • Perform an on-resonance Rx(q) rotation on every qubit to correct the spectator qubits’ rotations.

  24. Two-qubit Hamiltonian • Effective e-spin Hamiltonian in the rotating frame • Full Hamiltonian in the Lab. frame

  25. Exchange interaction J Strain • L. M. Kettle, H.-S. Goan, and S. C. Smith, PRB 73, 115205 (2006). See also:B. Koiller, X. Hu and S. Das Sarma, PRL 88, 072903 (2002).

  26. Two-qubit control • Two qubit Hamiltonian: The magnitude of the exchange interaction,J, depends on the degrees of overlap of electronic wave functions and can be controlled by the surface J-Gate. B. Kane, Nature 393, 133 (1998)

  27. Universal and CNOT gate • CNOT + single qubit rotations are universal for quantum computation. • Any gate can be constructed using CNOT and single qubit rotations. • What is the CNOT (Controlled-Not) gate: • Task is to demonstrate that the CNOT gate and single qubit rotations may be constructed.

  28. Constructing CNOT gate from the controlled Z Gate Hadamard gate: Controlled-Z gate, Controlled-Not gate:

  29. Construction of two-qubit gates • Any two-qubit gate may be expressed in the following way: • where W1, W2, W3 and W4 are local operations. We can perform these operations using single-qubit rotations. • The only challenge is to perform the entangling part of the gate. • What we have: • What we want: • Isolate the Z-Z term:

  30. Canonical decomposition of CNOT gate for global control e-spin QC CNOT gate operation time: 297ns C. D. Hill, L. C. L. Hollenberg, A. G. Fowler, C. J. Wellard, A. D. Greentree, and H.-S. Goan,“Global control and fast solid-state donor electron spin quantum computing”, Phys. Rev. B 72, 045350 (2005).

  31. Simulation of electron exchange mediated two-qubit gates in the Kane donor nuclear spin scheme showed that the gate fidelity is limited primary by the electron coherence when the electron dephasing timescale is close to the typical gate operation time of O(ms). • Experimental indication: P donor electron spin T2 > 60 msat4K in purified silicon [Tyryshkin, Lyon et al., PRB (2003)]. • Features of e-spin based QC: • Fast gate speed, • Comparatively simpler readout

  32. Optimal control • One of the important criteria for physical implementation of a practical quantum computer is to have a universal set of quantum gates with operation times much faster than the relevant decoherence time of the quantum computer. • High-fidelity quantum gates to meet the error threshold of about 10-4 (10-3) are also desired for fault-tolerant quantum computation (FTQC). • Thus the goal of optimal control is to find fast and high-fidelity quantum gates. Error threshold: P. Aliferis and J. Preskill, Phys. Rev. A 79, 012332 (2009).

  33. GRadient Ascent Pulse Engineering (GRAPE) • N. Khaneja et al., • J. Magn. Reson. • 172, 296 (2005). • A. Sporl et al., • Phys. Rev. A • 75, 012302 (2007) • Propagator during time step j (Dt=T/N) • Propagator at final time T See also: Montangero et al., PRL 99, 170501 (2007) and Carlini et al., PRL 96, 060503 (2006);PRA 75, 042308 (2007) • Performance function (fidelity) Nielsen et al., Science; PRA (2006) • Optimize the performance function (fidelity) w.r.t. the control amplitudes ukj in a given time T. • The minimum time sequence that meets the required error threshold is the near time-optimal control sequence.

  34. Trace Fidelity versus gate time Optimizer: spectral projected gradient method Stopping criteria of the error threshold : 10-9 30 piecewise constant steps is sufficient

  35. Choice of the value of Bac • While the target electron spin qubit will perform a particular unitary operation within time t, every spectator qubit will rotate around the x-axis with an angle of • If qx does not equal to 2np, where n is an integer, another correction step will be required for the spectator qubits. Therefore, it will be more convenient to choose the operation time,

  36. Near time-optimal control sequence 30 steps in100ns with an error of 1.11x10-16 Calculations performed using the effective e-spin Hamiltonian

  37. Canonical decomposition of CNOT gate for global control e-spin QC CNOT gate operation time: 297ns C. D. Hill, L. C. L. Hollenberg, A. G. Fowler, C. J. Wellard, A. D. Greentree, and H.-S. Goan,“Global control and fast solid-state donor electron spin quantum computing”, Phys. Rev. B 72, 045350 (2005).

  38. Parallel quantum computing • Traditional decomposition method that decomposes general gate operations into several single-qubit and some interaction (two-qubit) operations in series as the CNOT gate in the globally controlled electron spin scheme. So the single-qubit operations and two-qubit (interaction) operations do not act on the same qubits at the same time. • The GRAPE optimal control approach is in a sense more like parallel computing as single-qubit (A1 and A2 both on) and two-qubit (J on) operations can be performed simultaneously on the same qubits in parallel. • As a result, the more complex gate operation it is applied, the more time one may save, especially for those multiple-qubit gates that may not be simply decomposed by using the traditional method.

  39. Time evolution of the near time-optimal CNOT gate with input states |00> and |01> Simulations performed using the full Hamiltonian

  40. Time evolution of the near time-optimal CNOT gate with input states of |10> and |11> Simulations performed using the full Hamiltonian

  41. Summary of the CNOT gate fidelities • After about 60 (250) times of CNOT operations, the error sums up to 1.03x10-4 (0.79x10-4) and one has to reinitialize the nuclear spin state in order to maintain fault-tolerant quantm computation. • In the paper by A. Sporl et al., Phys. Rev. A 75 012302 (2007), TCNOT=55ps isabout 5 times faster than the pioneering experiment of coupled superconducting Josepson charge qubits [canonical decomposition]; with an error of 10-9 using the effective Hamiltonian, (when including higher charge states, the leakage is less than 1%).

  42. Control voltage fluctuations (noise) • Since we apply voltages on the A and J gates to control the strengths of hyperfine interaction and exchange interaction, there might be noise induced from the (thermal) fluctuations in the control circuits, which then cause the uncertainties of the control parameters and decrease the fidelity of a specific operation. • We model the noise on the control parameters A1, A2 and J as independent white noise with Hamiltonian

  43. Contour plot of logarithmic errors We simulate the optimal control sequence in the presence of the white noise through the effective master equation approach. • To satisfy the error threshold 10-4(10-3) of FTQC, the spectral densities, (GJ/h)2 and (GA/h)2 have to be smaller than 6.2Hz and 13Hz (63Hz and 125Hz), respectively. • This precision of control should be achievable with modern electronic voltage controller devices as the spectral density of energy fluctuations of the control parameters for good room temperature devices can be estimated to be 10-4~10-2Hz..

  44. Effect of decoherence • The decoherence time T2 for P donor electron spin in purified Si has been indicated experimentally to be potentially considerably longer than 60ms at 4K. • The error with decoherence can be estimated to be where Frand t are the trace fidelity and operation time of the gate, respectively. • For this simple estimate, the error is about 2.7x10-6, below the FTQC error threshold of 10-4 (10-3).

  45. Conclusions • A great advantage of the optimal control gate sequence is that the maximum exchange interaction is about 500 times smaller than the typical exchange interaction of J/h=10.2 GHz in the Kane’s originalproposal and yet the CNOT gate operation time is still 3 times faster than that in the globally controlled electron spin scheme. • This small exchange interaction relaxes significantly the stringent distance constraint of two neighboring donor atoms of 10-20nm as reported in the original Kane's proposal to about 30nm. To fabricate surface gates within such a distance is within reach of current fabrication technology. • Each step of the control sequence is about 3.3ns which may be achievable with modern electronics.

  46. Conclusions • The CNOT gate sequence we found is with high fidelity, above the fidelity threshold required for fault-tolerant quantum computation. • The fidelity of the gate sequence is shown, by using realistic (device) parameters, to be robust against control voltage fluctuations, electron spin decoherence and dipole-dipole interaction. • The GRAPE time-optimal control approach is in a sense more like parallel computing. The more complex gate operation it is applied, the more time one may save, especially for those multiple-qubit gates that may not be simply decomposed by using the traditional method. • The GRAPE technique may be proved useful in implementing (complex) quantum gate operations. • Ref: D.-B. Tsai, P.-W. Chen and H.-S. Goan, Phys. Rev. A 79, 060306 (Rapid Communication) (2009).

  47. Silicon-based electron-mediated nuclear spin quantum computer B. Kane, Nature (1998) • Exploiting the existing strength of Si technology • Qubits are nuclear spins of P donors in a regular array in pure silicon • Low temperature: • Effective Hamiltonian involves only spins • Long spin coherence and relaxation times • Magnetic field B to polarized electron spins • Control with surface gates and NMR pulses • Donor separation ~ 20nm • Gate width < 10nm

  48. Top-down approach for few qubit devices Controlled single-ion implantation • 14 KeV P ion beam is used to implant P dopants to an average depth of 15nm below the Si-SiO2 • Ion-stopping resist defines the array sites • Each ion entering the Si substrate produces e-hole pair that drift in an applied electric field • Created single current pulse for each ion strike is detected by on-chip single ion detector circuit. 95% confidence in ion detection 50% confidence in each 2-donor device

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