Quantum Control Synthesizing Robust Gates T. S. Mahesh Indian Institute of Science Education and Research, Pune

Contents DiVincenzo Criteria Quantum Control Single and Two-qubit control Control via Time-dependent Hamiltonians Progressive Optimization Gradient Ascent Practical Aspects Bounding within hardware limits Robustness Nonlinearity Summary

Criteria for Physical Realization of QIP Scalable physical system with mapping of qubits A method to initialize the system Big decoherence time to gate time Sufficient control of the system via time-dependent Hamiltonians (availability of a universal set of gates). 5. Efficient measurement of qubits DiVincenzo, Phys. Rev. A 1998

how best can we control its dynamics? Quantum Control Given a quantum system, how best can we control its dynamics? Control can be a general unitary or a state to state transfer (can also involve non-unitary processes: eg. changing purity) Control parameters must be within the hardware limits Control must be robust against the hardware errors Fast enough to minimize decoherence effects or combined with dynamical decoupling to suppress decoherence

General Unitary Hilbert Space 1 UTG UEXP 0 General unitary is state independent: Example: NOT, CNOT, Hadamard, etc. Hilbert Space 1 UTG UEXP obtained by simulation or process tomography 0 Fidelity =  Tr{UEXP·UTG} / N 2

State to State Transfer A particular input state is transferred to a particular output state Eg. 000  ( 000 + 111 ) /2 Hilbert Space Target Final obtained by tomography Initial Fidelity = FinalTarget 2

Universal Gates Local gates (eg. Ry(), Rz()) and CNOT gates together form a universal set Example: Error Correction Circuit Chiaverini et al, Nature 2004

Degree of control Fault-tolerant computation - E. Knill et al, Science 1998. Quantum gates need not be perfect Error correction can take care of imperfections For fault tolerant computation: Fidelity ~ 0.999

Single Qubit (spin-1/2) Control (up to a global phase) Bloch sphere

~ NMR spectrometer B0 B1cos(wrft) RF coil Pulse/Detect Sample resonance at 0 =B0 ~ B0 Superconducting coil B1cos(wrft)

~ Control Parameters B0 B1cos(wrft)  01 = 0 - ref 1 = B1  rf All frequencies are measured w.r.t. ref  RF offset =  = rf - ref  (kHz rad) Chemical Shift 01 = 0 - ref 1 = B1  rf ~ time B0  RF duration 1 RF amplitude  RF phase  RF offset B1cos(wrft)

Single Qubit (spin-1/2) Control (in RF frame) x (in REF frame) y Bloch sphere 90-x 90x y A general state: (up to a global phase)

Single Qubit (spin-1/2) Control (in RF frame) (in REF frame)

Single Qubit (spin-1/2) Control (in RF frame) (in REF frame) Turning OFF 0 : Refocusing y X Refocus Chemical Shift  time x w01

Two Qubit Control Local Gates

Qubit Selective Rotations - Homonuclear Band-width  1/   = 1 1 2 dibromothiophene non-selective  = 1 selective Not a good method: ignores the time evolution

~ Qubit Selective Rotations - Heteronuclear 13CHCl3 1H (500 MHz @ 11T) 13C (125 MHz @ 11T) Larmor frequencies are separated by MHz Usually irradiated by different coils in the probe No overlap in bandwidths at all Easy to rotate selectively ~

Two Qubit Control Local Gates CNOT Gate

Two Qubit Control   Refocussing: X 1 2   Z X 1 2  Chemical shift Coupling constant Refocussing: X Refocus Chemical Shifts 1 2   Z Rz(90)   = 1/(4J) time X Refocus 0 & J-coupling 1 2  Rz(0)  time

Two Qubit Control Z H = = Chemical shift Chemical shift Coupling constant Z H = 1/(4J) R-z(90) time X R-y(90) =

Control via Time-dependent Hamiltonians H = H (a (t), b (t) , g (t) , ) a (t) t NOT EASY !! (exception: periodic dependence)

Control via Piecewise Continuous Hamiltonians b3 g3 H 3 a1 b1 g1 H 1 a2 b2 g2 H 2 a4 b4 g4 H 4 Time

Numerical Approaches for Control Progressive Optimization D. G. Cory & co-workers, JCP 2002 Mahesh & Suter, PRA 2006 Gradient Ascent Navin Khaneja et al, JMR 2005 Common features Generate piecewise continuous Hamiltonians Start from a random guess, iteratively proceed Good solution not guaranteed Multiple solutions may exist No global optimization

Piecewise Continuous Control D. G. Cory, JCP 2002 Strongly Modulated Pulse (SMP) (t3,w13,f3,w3) … (t1,w11,f1,w1) (t2,w12,f2,w2)

Progressive Optimization D. G. Cory, JCP 2002 Random Guess Maximize Fidelity simplex Split Maximize Fidelity simplex Split Maximize Fidelity simplex

Example Fidelity : 0.99 Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : (/2)y1 Fidelity : 0.99

Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : (/2)y1

Initial state Iz1+Iz2 SMPs are not limited by bandwidth Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : (/2)y1 Initial state Iz1+Iz2 SMPs are not limited by bandwidth

Initial state Iz1+Iz2 SMPs are not limited by bandwidth Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : (/2)y1 Initial state Iz1+Iz2 SMPs are not limited by bandwidth

1 2 3 0.99 0.99 0.99 CH3 C NH3+ O -O H 3 1 2 13C Alanine Amp (kHz) Pha (deg) 0.99 Amp (kHz) Pha (deg) 0.99 Amp (kHz) Pha (deg) CH3 C NH3+ O -O H 3 1 2 Time (ms) 13C Alanine

Shifts and J-couplings AB 1 2 3 4 5 6 7 8 9 10 11 12 -1423 134 6.6 -13874 52 35.2 4.1 2.0 1.8 5.3 1444 2.2 74 11.5 4.4 -9688 53.6 147 6.1 201 8233 998 3.6 4.3 6.7 -998 4421 16.2 4279 2455 221.8 1756 -3878

Benchmarking 12-qubits A 8 A’ 2 11 1 3 10 5 4 9 7 6 Qubits Time Benchmarking circuit AA’ 1 2 3 4 5 6 7 8 9 10 11 Qubits Time Fidelity: 0.8 PRL, 2006

Quantum Algorithm for NGE (QNGE) : in liquid crystal PRA, 2006

Quantum Algorithm for NGE (QNGE) : Quantum Algorithm for NGE (QNGE) : Crob: 0.98 PRA, 2006

Progressive Optimization D. G. Cory, JCP 2002 Advantages Works well for small number of qubits ( < 5 ) Can be combined with other optimizations (genetic algorithm etc) Solutions consist of small number of segments – easy to analyze Disadvantage 1. Maximization is usually via Simplex algorithms Takes a long time

SMPs : Calculation Time During SMP calculation: U = exp(-iHeff t) calculated typically over 103 times Qubits Calc. time 1 - 3 minutes 4 - 6 Hours > 7 Days (estimation) Single ½ : Heff = 2 x 2 Two spins : Heff = 4 x 4 . Matrix Exponentiation is a difficult job - Several dubious ways !! 210 x 210 ~ Million 10 spins : Heff =

Gradient Ascent Final density matrix: Navin Khaneja et al, JMR 2005 Liouville von-Neuman eqn Control parameters Final density matrix:

Gradient Ascent Navin Khaneja et al, JMR 2005 Correlation: Backward propagated opeartor at t = jt Forward propagated opeartor at t = jt

Gradient Ascent ’  = ’ t ? (up to 1st order in t) Navin Khaneja et al, JMR 2005  = ’ t ’ ? (up to 1st order in t)

Gradient Ascent Navin Khaneja et al, JMR 2005 Step-size

Gradient Ascent GRAPE Algorithm Stop Guess uk Navin Khaneja et al, JMR 2005 GRAPE Algorithm Guess uk No Correlation > 0.99? Yes Stop

Practical Aspects Bounding within hardware limits Robustness Nonlinearity

Shoots-up if any control parameter exceeds the limit Bounding the control parameters Quality factor = Fidelity + Penalty function Shoots-up if any control parameter exceeds the limit To be maximized

Practical Aspects Bounding within hardware limits Robustness Nonlinearity

Incoherent Processes Spatial inhomogeneities in RF / Static field Hilbert Space Final Final Final Initial UEXPk(t)

Robust Control Coherent control in the presence of incoherence: Hilbert Space Target Initial UEXPk(t)

Inhomogeneities SFI Analysis of spectral line shapes RFI Analysis of nutation decay Ideal SFI f f z z Ideal RFI y y x x

RF inhomogeneity 1 Ideal Probability of distribution In practice RFI: Spatial nonuniformity in RF power RF Power Desired RF Power

RF inhomogeneity Bruker PAQXI probe (500 MHz)

Example Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : (/2)y1

Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : (/2)y1

Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : (/2)y1

Initial state Iz1+Iz2 Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : (/2)y1 Initial state Iz1+Iz2

Initial state Iz1+Iz2 Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : (/2)y1 Initial state Iz1+Iz2

Robust Control Initial state Ix1+Ix2 - Eg. Two-qubit system Shifts: 500 Hz, -500 Hz J = 50 Hz Fidelity = 0.99 Target Operator : ()y1 Initial state Ix1+Ix2 -

Robust Control - Initial state Ix1+Ix2 Eg. Two-qubit system Shifts: 500 Hz, -500 Hz J = 50 Hz Fidelity = 0.99 Target Operator : ()y1 - Initial state Ix1+Ix2

Practical Aspects Bounding within hardware limits Robustness Nonlinearity

Spectrometer non-linearities Computer: “This is what I sent”

Spectrometer non-linearities Spins: “This is what we got” Computer: “This is what I sent”

~ Multi-channel probes: Target coil Spy coil - D. G. Cory et al, PRA 2003.

Spectrometer non-linearities F

Feedback correction F hardware F-1 F hardware - D. G. Cory et al, PRA 2003.

Feedback correction: Computer: Spins: “This is what we got” “This is what I sent” Spins: “This is what we got” Compensated Shape - D. G. Cory et al, PRA 2003.

Summary DiVincenzo Criteria Quantum Control Single and Two-qubit control Control via Time-dependent Hamiltonians Progressive Optimization Gradient Ascent Practical Aspects Bounding within hardware limits Robustness Nonlinearity