Optimal gates in imperfect qubits

Optimal gates in imperfect Learning to live with decoherence 1 2 M.J. Storcz 2 J. Ferber 2 A. Spörl 3 T. Schulte-Herbrüggen 3 S.J. Glaser 3 P. Rebentrost 12 1 Institute for Quantum Computing (IQC) and Department of Physics and Astronomy University of Waterloo, Canada 2 Physics Department, Arnold Sommerfeld Center, and CeNS Ludwig-Maximilians-Universität München, Germany 3 Chemistry Department Munich University of Technology, Germany DISQ conference, UC Berkeley

Contents 1 is not a constant 2 Quantum logic in the presence of a two-level fluctuator 3 Avoiding leakage in one-qubit gates

Generalized master equations The phenomenological Bloch equation Ṁ z = γ( B M) z + (M z M eq )/T 1 Ṁ x/y = γ( B M) x/y + M x/y /T 2. Low T : T 1 and T 2 not constant. 1 = cos 2 θs(e) T 1 1 = 1 + sin 2 θs(0) T 2 2T 1 Noise power S(ω), angle θ between and B: Hamiltonian and relaxation depend on parameters u ρ = i[h u, ρ] + D u (ρ) (1)

Slow fluctuators Ĥ S = E 1 (t)ˆσ z + ˆσ x + Λˆσ z X(t) X(t)X(0) ω 1/ω

Noise physics I Random telegraph Lorentzian power spectrum Superposition of many leads to 1/f.

Noise on the Bloch sphere Change of precession frequency No transition 1/T 1 1/T 2 = S(0) Low-frequency power (high) No change of precession B / B z = O(B z /B) Environment coupling needs transition 1/T 1 = S(B) 1/T 2 = 1/2T 1 High-frequency power (low)

Optimum working point Optimum working point: E (Vion et al, 2002) σ x gate from free precession σ z gate from low-amplitude Rabi s,e 1 (t) = ω r cos t, ω r Initial success: Push T 2 from 20ns to 500ns. Combine with Echo (Bertet, 2005): Go up to 5µs Questions: is this a true optimum? how can we generalize this to larger amplitude? Duration t g π/e 1 π/ = t Ramsey Can we do fast optimum working point s?

Spinning during Rabi s: Spin locks to the driving frequency Ω Rotating frame S(ω) rot S(ω + Ω), i.e. 1 1 T Rabi T S(Ω). (2) 2 rot Spinning averages over slow fluctuators.

Spin echo: Average out time ensemble inhomogeniety. Nakamura, PRL 2002. Bertet, PRL 2005 Dominated by single fluctuator. (Galperin, 2004).

Composite s Repeated spin echo (bang-bang) stabilizes quantum memory. (Faoro, 2004; Gutmann, 2005) For gates: Composite s. i.e. compact CORPSE sequence (Collin, PRL 2004)

Other composite s Similar for two-qubit gates: FLICFORQ, parametric coupler etc. Problem: Tedious construction by hand / NMR analogies. Issue: How to best find the right NMR-style trick? Experience? Encyclopedia?

Optimal control Take any dynamical system with variables x i and controls u j with EOM ẋ = f (x, u, t) (3) Optimize a performance index at final time t f, φ(x(t f ), u(t f )) using J = φ(x(t f ), u(t f )) + (4) tf t i dtλ T (t)(ẋ f (x, u, t)) with initial conditions x(t i ).

Optimizing Pulse Shapes Control problem for a quantum gate: x U(t) U(t i ) = ˆ1 f i(h d + u i (t)h i )U i φ = U gate U(t f ) 2 = 2N 2ReTr(U gateu(t f )) So we need to maximize Tr(U gateu(t f )). Problem: Fixes global phase, too Solution: Maximize Φ = Tr(U gateu(t f )) 2 instead.

Challenge In the discretized grid, how does Φ change when the control is changed in one point?

We can derive Φ u k Gradient Ascent Pulse Engineering (GRAPE) analytically. Φ ) = δtre [(TrU u j+1... U N U gateh k U j... U 1 k )] (TrU 1... U j U gateu N... U j+1 [N. Khaneja, JMR 05]

opengrape Open system dynamics: Master equation ρ(t) = ( ih(u(t)) + Γ(u(t)) ) ρ(t) (5) H = Ad H : Commutator with Hamiltonian Γ: Relaxation Formal solution: Linear quantum map F(t) F(0) = ˆ1, target map F gate = U Ū. Schulte-Herbrüggen, 2006 ρ(t) = F(t)ρ(0) (6)

Ĥ S = E 1 (t)ˆσ z + ˆσ x + E 2ˆτ z + Λˆσ z ˆτ z + i λ i (ˆτ +ˆbi + ˆτ ˆb i ) + i ω i b i b i. σ: Charge or flux qubit τ: two-level defect b: Heat bath, Ohmic spectrum J(ω) = i λ2 i δ(ω ω i) = κωθ(ω ω c )

Quantum of the model Perturbative picture: Quantum seen by with H Noise = E 1 (t)ˆσ z + (t)ˆσ x + F(t)ˆσ z (7) S RTN (ω) = F(t)F (0) ω = Λ2 π γ γ 2 + ω 2 (8) Peaked at low ω: Non-Markovian, non-perturbative Perturbative estimates using E = E1 2 + 2 : 1 = 2 Λ 2 γ T 1 E 2 πe 2 (9) 1 = 1 + E 1 2 Λ 2 T 2 2T 1 E 2 πγ 1 T 1 (10)

At the optimum point Optimal working point strategy: E 1 = 0 produces σ x weak Rabi E 1 = ω R cos t, duration t G = π/ω R produces σ z Is this truly optimal? Can we go beyond perturbation theory?

Master equation + control problem Qubit master equation non-markovian and strongly coupled. Complex environments approach - keep the impurity ˆρ S (t) = 1 i [ĤS, ˆρ S (t)] + [ˆσ + 1, ˆΣ 1 ˆρ S(t)] + [ˆσ 1, ˆΣ + 0 ˆρ S(t)] [ˆσ 1, ˆρ S(t)ˆΣ + 1 ] [ˆσ+ 1, ˆρ S(t)ˆΣ 0 ] (11) with rates ˆΣ ± s = 1 (i ) 2 0 dt 0 dωj(ω)(n(ω)+s)e ±iωt ˆτ ± (t ) (12) Control problem still on qubit alone. Trace over impurity after the evolution F R = tr TLF F(ρ eq TLF ) (13)

σ z gate execution time T Rebentrost, quant-ph/0612165

How good is good? Upper panel: Compare to 1 e t/t 1 and 1 e t/2t 1 Lower panel: Compare to rectangular Rabi E 1 = (π/t ) cos(2 t)

E 1 (!) Magic times 2 1 0 1>!1!> +>!2 0>!3 0 1 2 Time (1/!) 3 Self-refocusing: Control off most of time

Gate error 0.4 0.3 0.2 0.1 0.1 0.05 Dependence on bath parameters T=0.04" T=0.06" T=0.2" T=0.5" T=1.0" c+d/! Gate error 0 0.02 0.04! (") Maximal error 0.4 0.2 0 0 0.2 0.5 1 Temperature (") 0 1 2 3 4! (") Flipping rate: γ = 2κE 2 coth( E 2 /kt ) Low γ: Only inhomogeniety, high γ: Motional narrowing.

Related work Similar work: Möttönnen, PRA 2005; Ojanen, PRA 2007; Grace, J. Phys., 2007; Montangero PRL 2007.

Weak nonlinearities These are (by far) no strict two-level systems (as all ). Extreme case: Phase qubit, Transmon δω = ω 01 ω 12 0.1ω 01 Drive resonantly on ω 01 Fast gate large λ leakage to the higher level Hamiltonian ω 01 + ω 21 2λ(t) cos ω01 t 0 H = 2λ(t) cos ω01 t ω 01 λ(t) cos ω 01 t 0 λ(t) cos ω 01 t 0

Weak nonlinearities These are (by far) no strict two-level systems (as all ). Extreme case: Phase qubit, Transmon δω = ω 01 ω 12 0.1ω 01 Drive resonantly on ω 01 Fast gate large λ leakage to the higher level RWA Hamiltonian H = δω 2λ(t) 0 2λ(t) 0 λ(t) 0 λ(t) 0

NMR-style decoupling I Perturbation theory in λ/δω decouples transitions ( ) 0 1 Working transition 0 1: H work = λ 1 0 ( ) δω 2λ transition 1 2: H leak = 2λ 0 Composite λ π/8 0 π/4 0 π/8 ( W R π ) ( x 8 ˆ1 R π ) ( x 4 ˆ1 R π ) ( ) ( ) x ( 8 ) L R φ θ 8 R z (π) R φ θ 4 R z (π) R φ θ 8 R x (π) ( R z (φ ) + O λ 2 δω Tilted axis: σ θ (δω/2)σ z + λσ x Echo: R z (π)r θ (φ/4)r z (π) = R θ (φ/4).

Composite performance Pulse sequence already found by Steffen et al, 2003. Leaves allowable arbitrary phase on leakage level Supressed transitions to order λ/δω Duration t g = 2π/δω + t Rabi (two drifts plus Rabi rotations) Challenge: Faster + decouple to all orders in λ/δω.

Optimal shapes Minimize the Rabi time. Optimal time: t g δf = 1 + ɛ U gate = e iφ 1 e iφ 2 0 0 0 0 1 0 1 0 (14) Shelving of population in higher state.

More accessible shapes Slightly longer time leads to smooth Use penalties to prevent initial rise P. Rebentrost, T. Schulte-Herbrüggen, FKW, in preparation

Global performace Performance index φ = (U X U) 00 + (U X U) 11 2 Gaussian s Limit bandwidth, reduce drive at leakage transition Shortest shelving tδf = 1 Higher level = speed limit

H drift = δω 0 0 0 0 0 0 0 0 Why 2π and not π? 0 H control = 2λ(t) 0 2λ(t) 0 λ(t) 0 λ(t) 0 The simpler single echo sequence R x (π/2)r wait (π)r x (π/2) could be executed in half the time. π/2 + π/2 = π: rotates the working transition π/2 π/2 = 0: refocuses the 1 2 leakage transition so why does it not work? High Rabi s λ/δω > 1: generate 0 2 leakage transition Physics: Higher order transitions from high drive Math: Lie algebra generated by H drift and H control Three transitions, need three s to satifsy all constraints.

Optimal control finds s Find the irreducible part of the decoherence model Fast s at the optimum point Avoiding leakage Outlook Make it work in experiment! Two-qubit gates at the optimum point Tolerating inhomogeniety