Quantum Control of Qubits - PDF Free Download

Quantum Control of Qubits K. Birgitta Whaley University of California, Berkeley J. Zhang J. Vala S. Sastry M. Mottonen R. desousa

Quantum Circuit model input 1> 6> = 1> 0> H S T H S H x x 7 k = 0 e 3 π ik /2 k output e.g., Quantum Fourier Transform on 3 qubits wire = carrier of quantum information (qubit { 0>, 1>}, qudit { 0>, 1>,., d>} gate = time evolution of quantum information U U = T exp ( ih( t) t / ) 0

U Universal sets of quantum gates Theorem: every n-qubit unitary can be decomposed into combinations of 1-qubit and 2-qubit operations (Barenco et al, 1995) 1. single qubit gates iφ U(, nˆ, ) = e cos I + isin nˆ ( ) θ φ θ θ σ SU(2), rotations on Bloch sphere 2. Two qubit gates e.g., CNOT CNOT 00 01 10 11 1 0 0 0 0 1 0 0 = 0 0 0 1 0 0 1 0

Quantum simulators What can we implement, given a physical system? atoms in OL e.g., CPHASE SWAP PRL 91, 027903 (2003) interacting quantum spins controlled collisions spin-spin exchange What control fields are required? What cost? Can we simply generate arbitrary quantum operations?

QIP requires ultra-high level of quantum control high fidelity quantum operations required for faulttolerant quantum computation in standard model admissible error threshold - generic threshold value ~.0001 [Aharonov, Gottesman 02] - scaling: #levels recursion #qubits #operations 2 50 20000 3 350 4000000 experimental fidelities ~.01 need #qubits, #operations 1-3 orders of magnitude larger [ Roadmap Goal: recursion level 2 by 2012 ]

quantum control and robustness How generate gates and arbitrary quantum operations from Hamiltonians? Efficiency various criteria for optimality Time On/off switching of interactions and external fields Energy input from external fields Minimal decoherence All of the above together, with accurate gates.?

Algebraic approach Tunable interactions: 2-qubit gates by Weyl chamber steering Non-tunable interactions: algebraic decoupling for 1-qubit gates allows some gate optimization add optimal control optimize with respect to cost function -time -energy - decoherence

all 1- and 2-qubit gates: SU(4) schematic partition of SU(4) Local gates SU(2) SU(2) Non-local gates SU(4)\SU(2) SU(2) CNOT SWAP Perfect Entanglers SWAP 1/2 local algebra su(4): σ 1 i, σ2 i, σ 1 i σ2 j, i, j = x, y, z Abelian subalgebra: σ 1 x σ2 x, σ1 y σ2 y, σ1 z σ2 z Cartan decomposition U = k 1 A k 2 non-local

su(4) algebra = k p i k = span {σ x1, σ y1, σ z1, σ x2, σ y2, σ z2 } 2 local =( 0 1 σ ) x 1 0 σ y = ( 0 -i ) i 0 σ z = ( 1 0 ) 0-1 non-local i p = span {σ x1 σ x2, σ x1 σ y2, σ x1 σ z2, σ y1 σ x2, σ y1 σ y2, σ y1 σ z2, σ z1 σ x2, σ z1 σ y2, σ z1 σ z2 } 2 σ x1 σ x2 = ( 0 σ 2 x ) σ x 2 0 = 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 Commutation relation, e.g.: [σ y2, σ y1 σ z2 ] ~ - σ y1 σ x 2

Cartan decomposition: i k = span {σ x1, σ y1, σ z1, σ x2, σ y2, σ z2 } 2 i p = span {σ x1 σ x2, σ x1 σ y2, σ x1 σ z2, σ y1 σ x2, σ y1 σ y2, σ y1 σ z2, σ z1 σ x2, σ z1 σ y2, σ z1 σ z2 } 2 Maximal Abelian subalgebra a = span i {σ x1 σ x2, σ y1 σ y2, σ z1 σ z2 } 2 su(4) algebra = k p Decomposition of a unitary transformation U in SU(4) i U = k 1 A k 2 = k 1 exp[ (c 2 1 σ x1 σ x2 + c 2 σ y1 σ y2 + c 3 σ z1 σ z2 )] k 2 local local 3-Torus non-local

Local gates and local equivalence (~) U 1 ~ U 2 if U 1 = k 1 U 2 k 2, where k 1 and k 2 are local gates, e.g., CNOT ~ C-z SWAP ~ CNOT local equivalence can be determined by evaluating 3 invariants (Makhlin quant-ph/0002045)

Makhlin s local invariants Given a two-qubit operation U G 1 : complex number G 2 : real number 3 invariants Local gates CNOT SWAP G 1 1 0-1 i/4 G 2 3 1-3 0 Claim: If G 1 (U 1 )=G 1 (U 2 ) and G 2 (U 1 )=G 2 (U 2 ), then U 1 ~ U 2 Makhlin, QIP 1, 243 (2002)

Cartan decomposition on su(4) any U SU(4) can be decomposed as: invariants are periodic in c 1, c 2, c 3

Geometric Theory of Non-local Gates Cartan decomposition U = k 1 A k 2 = k 1 exp[ i (c 1 σ x1 σ x2 + c 2 σ y1 σ y2 + c 3 σ z1 σ z2 )] k 2 2 c 1, c 2, c 3 periodic 3-Torus J. Zhang et al., PRA 67, 042313 Local invariants (2003) G 1 = cos c 1 cos c 2 cos c 3 G 2 = sin c 1 sin c 2 sin c 3 G 3 = 2 (cos 2 c 1 + cos 2 c 2 + cos 2 c 3 ) - 3 G 1,G 2 and G 3 are invariant on permuting c 1, c 2, and c 3 with/without sign flips cartesian representation symmetry reduction Weyl tetrahedron B SWAP one-to-one correspondence between the points inside the tetrahedron and local equivalence classes (except on base) CNOT DCNOT

Implications of geometric analysis Tetrahedral representation of local equivalence classes Applications: physical generation of non-local gates, arbitrary 2- qubit operations optimally efficient quantum circuits characterization of perfect entanglers

Generation of non-local gates as a steering problem in the Weyl tetrahedron 15 dimensional control problem on U(4) reduction SWAP CNOT 3 dimensional steering problem in Weyl tetrahedron quotient space

Weyl tetrahedron trajectory System dynamics: For any t, determines a point in the tetrahedron via the Makhlin invariants for the non-local equivalence classes, i.e., U() t G () t c () t i i

Weyl tetrahedron trajectory I (t=0)

Weyl tetrahedron trajectory I (t=0) U 1 (t=t 1 )

Weyl tetrahedron trajectory U 2 (t=t 2 ) I (t=0) U 1 (t=t 1 )

Weyl tetrahedron trajectory U 2 (t=t 2 ) As time evolves, we can obtain a continuous trajectory in the Weyl tetrahedron I (t=0) U 1 (t=t 1 )

Pure nonlocal Hamiltonian Consider [c x, c y, c z ]

Pure nonlocal Hamiltonian Consider [c x, c y, c z ] [c x, -c y, -c z ] [-c x, c z, -c y ], where k Weyl group Reflections of [c x, c y, c z ] w.r.t. diagonal planes New directions: [c x, -c y, -c z ], [-c x, c z, -c y ], [c x, c z, c y ], [-c x, -c z, c y ],

Steering a Weyl chamber trajectory Piece two segments together: we can reach anywhere in the plane spanned by [c x, c y, c z ] and [c x, -c y, -c z ]. Changing direction twice suffices: the trajectory defines a quantum circuit Theorem: the following circuit can implement any two-qubit gate

Minimum bound for Controlled-Unitary For a Controlled-U gate, minimum applications needed to implement any arbitrary two-qubit gate is. CNOT The most efficient among Controlled-U J. Zhang et al, PRA 69 042309 (2004)

Quantum Circuits for arbitrary 2-qubit operations CNOT 3 applications suffice SWAP CNOT Double-CNOT 3 applications suffice SWAP DCNOT J. Zhang, J. Vala, S.Sastry, K.B. Whaley PRA 69, 042309 (2004)

J. Zhang et al., PRL 93, 020502 (2004) B = (π/2, π/4,0) Two applications of the B gate suffice to implement any arbitrary two-qubit gate: explicit solutions for β 1 and β 2 as functions of c 2 and c 3 on the computational basis B acts as:

Example: SWAP gate and quantum wire qubit state transfer via a sequence of SWAP gates: Ψ> 0> 0> 0> 0> 1 2 3 time n-1 n Oskin et al., Proc. ISCA 2002 SWAP [(c 0 0> k +c 1 1> k ) 0> k+1 ] = 0> k (c 0 0> k+1 +c 1 1> k+1 ) only two B gates needed compared to three CNOTs for each step of the wire 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 B SWAP

Example: QFT on two qubits computer science implementation of two-qubit QFT: FIVE CNOT gates in total in matrix representation: three CNOT gates can implement this * : ~ B gate needs only two applications: in n-qubit case, the B gate is slightly better than the CNOT gate * J. Zhang et al., quant-ph/0308167

I. Josephson junction charge-coupled qubits Y. Makhlin et al., RMP 73, 357 (2001) tune BVt ( () ) z g E( Φt) J x () charge qubit with tunable coupling: for 2 qubits 1 1 H B E EE 1 2 i i i J J 1 2 zσ z Jσx σ yσ y i= 1,2 2 i= 1,2 2 EL = i 1 2 switch E J independently switch EJ, EJ together 1 qubit operations 2 qubit operations E 100 mk, E 1 100mK J L

interaction between Josephson junction qubits: H int = -(αe L /2) (σ x1 + σ x2 ) + α 2 E L σ y 1 σ y 2 (α=e J /E L ) curvature translation Weyl chamber trajectory perfect entanglers [CNOT] time optimal parameters for CNOT, α = 1.1991, t=2.73 J. Zhang et al., PRA 67, 042313 (2003)

scaled parameters E J =αe L, E L =1: tune α to implement various gates in minimum time CNOT B α > 1 time optimal solution for CNOT has α > 1 no CNOT solution for α <1 B gate has solution for all α regimes B realistic SC circuit, α 1 - no CNOT from single application of H int - but B can be implemented directly α < 1 Zhang et al. PRL 93, 020502 (2004)

II. inductively coupled SC flux qubits single flux qubit H 1 = ε () t σ +Δσ 2 () i () i i i z i x Φ /2 a =Φ0 Flux qubit: Chiorescu et al. Science (2003)

inductive coupling: natural interaction via flux: e.g., screening flux of qubit 1 changes flux bias ε of qubit 2 σ z (1) σ z (2) interaction 1 ( i i H = ε () t σ +Δ σ ) + Kσ σ 2 i= 1,2 () () (1) (2) i z i x z z coupling via mutual inductance: K fixed new - magnetic flux J in the outer loop couples the two qubits: K tunable and can be switched off B. Plourde, T. Robinson, F. Wilhelm, J. Clarke, et al.

Entangling operation with variable inductance Switch magnetic flux in outer loop on/off with bias current, couples two qubits 1 ( i i H = ε ε () t σ +Δ σ ) + K() t σ σ 2 i= 1,2 () () (1) (2) i z i x z z ε t = ε + A ωt+ φ + δε t (0) xtalk i() i icos( i i) i () 2-qubit operations: Weyl trajectory 1-qubit operations: external control fields ω i (off resonant)

Implementation of CNOT CNOT scale up B. Plourde et al, PRB 70 140501 (2004)

Non-tunable interactions: how generate 1-qubit gates? direct approach: exact algebraic decoupling of two-qubit Hamiltonian J is the always-on and untunable coupling strength, ω j and φ j and phases of the external control fields the amplitudes Target: generate any arbitrary single-qubit operation in each qubit J. Zhang and K. B. Whaley, PRA 73 022306 (2006)

Simplified problem It is easy to prove that to implement any arbitrary one-qubit operation, we only need to generate an arbitrary local unitary operation: from the Hamiltonian: Now observe that iσ x1 /2, iσ x 2/2, and iσ z1 σ z2 /2 generate the following Lie algebra: It is straightforward to show that satisfies the same commutation relations as so(4), where so(4) denotes the Lie algebra formed by all the 4x4 real skew symmetric matrices.

Lie algebra isomorphism Let we have the following commutation relations: Therefore, is isomorphic to. This isomorphism allows simplification for the generation of single-qubit operation, because it provides an algebraic way to decouple the entangling Hamiltonian into two unentangled single-qubit Hamiltonians.

Two sub-problems We can now rewrite the Hamiltonian as and the target operation as Now the original problem of generating from becomes generating from We only need to implement the following two one-qubit operations: (1) Generate from the Hamiltonian ; and (2) Generate from the Hamiltonian.

One-qubit sub-operations: optimal control Consider a general one-qubit system: Solution of ω(t) to exactly implement a target one-qubit operation is possible with simultaneous minimization of a cost function Time optimal Energy optimal J T = 1dt J 0 T 1 2 = ω 2 0 () tdt Also have analytic approximate implementation with γ J ω = cos ( Jt), T = 2 nπ / J nπ

Simple example: Let J=200 Hz in the Hamiltonian and be the desired target 1-qubit operation. Choosing and n=1, we obtain an approximate solution The corresponding pulse time is T=31.4 ms. Numerical optimization via the maximization of the fidelity leads to the improved solution parameters with corresponding pulse time T=31.911 ms and fidelity error 4.104 x 10-11. Control functions that generate. (A) Dashed line: approximate control; solid line: fidelity optimized control; (B) the difference between fidelity optimized control and minimum energy control

Optimal control of 1-qubit operations subject to random telegraph noise 0 H = a( t) σ +η() t σ x z Bounded control a a max Random telegraph noise 1 Solution for given noise correlation time via numerical optimization of unitary quantum trajectory formulation of fidelity Noise! M. Möttönen, R. de Sousa, J. Zhang, K. B. Whaley, PRA 73, 022332 (2006)

Summary of Part I Geometric approach to non-local gates - steering approach to generation of 2-qubit gates - analytic construction of quantum circuits How implement an arbitrary 2-qubit operation? - starting from given Hamiltonian steering in Weyl chamber (tetrahedron) - starting from given gate, e.g., CNOT gate B is optimal, only 2 applications Constraints - physical feasibility of H - minimal switchings, time optimization Algebraic decoupling for 1-qubit operations when non-tunable interactions present - optimization with respect to general cost function - time, energy, decoherence broad route to optimal feasible control of coupled qubits

Part II Geometric approach to non-local gates - steering approach to generation of 2-qubit gates - analytic construction of quantum circuits How implement an arbitrary 2-qubit operation? - starting from given Hamiltonian steering in Weyl chamber (tetrahedron) - starting from given gate, e.g., CNOT gate B is optimal, only 2 applications Constraints - physical feasibility of H - minimal switchings, time optimization Algebraic decoupling for 1-qubit operations when non-tunable interactions present - optimization with respect to general cost function - time, energy, decoherence broad route to optimal feasible control of coupled qubits

Random telegraph noise Average sojourn τ c η(t) ξ(t) 1.0 0.5 ~ S(ω) 1.0 0.8 Lorentzian spectrum 0.0 time 0 2 4 6 8 10 0.6 0.4 Λ cut-off =1/τ c -0.5 0.2-1.0 0.0 0 1 2 3 4 5 6 ωτ c

Random telegraph noise (RTN) Described by the correlation time τ c and the noise strength Δ The noise amplitude jumps between values Δ and Δ Probability of no jumps in time t is Jump time instants and the noise amplitude are given by

Trapping center noise on charge qubits S λ (ω) / ( λ 2 /γ) ~ 0.25 0.20 0.15 0.10 0.05 T / γ >> 1 T / γ << 1 0.00-3 -2-1 0 1 2 3 ω / γ ω/γ Effective only if trap energy level is close to Fermi level; High temperature, k B T>>γ: Lorentzian spectrum, semiclassical RTN; Low temperature, k B T<<γ : QUANTUM REGIME f-noise, Ohmic R. de Sousa, K.B. Whaley, F.K. Wilhelm, J. von Delft, PRL 95, 247006 (2005)

System dynamics Let k index the sample paths of RTN The dynamics of the system density matrix is given by an average over all different noise samples as

Example operations Bit flip 0> 1> NOT gate Fidelity gate = target gate U t Fidelity

Gradient ascent pulse engineering (GRAPE) Optimizes the fidelity with respect to the control pulse by a gradient method Solution not unique We used a constant control pulse as an initial condition Convergence of fidelity much faster than convergence of the pulse shape Compare with standard pulse sequences for correction of systematic (static) error, CORPSE and SCORPSE

Composite pulse sequences π-pulse: Compensation of off-resonence with a pulse sequence CORPSE: short CORPSE: Best short pulse sequences correcting for systematic static error

CORPSE and short CORPSE 1 0.5 a/a max 0 0.5 1 0 2 4 6 t 8 10 12 14

Composite pulses and numerical optimization CORPSE and SCORPSE: Cummins, Llewellyn, Jones, PRA 67, 042308 (2003) U π π i + σ x + ησ z 2 η = e Fidelity 1 amax 2 U CORPSE 7π 5π π 4 i + σx+ ησ 6 z i σx+ ησ z i + σ x ησ z 6 + 6 η = e e e Fidelity 1 0.065 amax U Short CORPSE 5 4 i π π π σx ησ 6 z i σx ησ + + + 6 z i σx+ ησ 6 z η = e e e Fidelity 1 2.7 amax Large number of composite pulses Numerical optimization (Gradient Ascent Pulse Engineering) N. Khaneja et al, J. Magn. Reson. 172, 296 (2005)

Fidelity vs noise correlation time for the state transformation 0> 1> GRAPE short CORPSE Fidelity CORPSE π-pulse

Fidelity vs noise correlation time for NOT gate GRAPE Fidelity short CORPSE π-pulse CORPSE

Optimized operation times Bit flip 0> 1> Δ=0.25 a max =5 NOT gate Δ=0.125 a max =30

OVERALL SUMMARY: High fidelity quantum gate operations from Hamiltonians Weyl chamber steering for 2-qubit (non-local) gates Algebraic decoupling for 1-qubit (local) gates in presence of untunable interactions Efficiency issues implementation specific Decoherence suppression using bounded controls for broad range of noise correlation times current/future: combined methodologies for optimal feasible quantum control tailored to specific qubit systems