Lie algebraic aspects of quantum control in interacting spin-1/2 (qubit) chains

Transcription

1 .. Lie algebraic aspects of quantum control in interacting spin-1/2 (qubit) chains Vladimir M. Stojanović Condensed Matter Theory Group HARVARD UNIVERSITY September 16, 2014 V. M. Stojanović (Harvard) TI CMSA September 16, / 25

2 . Outline of the talk Introduction to quantum (coherent) control Local quantum control of Heisenberg spin-1/2 chains R. Heule, C. Bruder, D. Burgarth, and VMS, PRA 82, (2010); Eur. Phys. J. D 63, 41 (2011). VMS, A. Fedorov, A. Wallraff, and C. Bruder, PRB 85, (2012). Pulse-sequence generated dynamics of stabilizer operators: application in measurement-based QC, i.e., one-way QC T. Tanamoto, D. Becker, VMS, and C. Bruder, PRA 86, (2012); T. Tanamoto, VMS, C. Bruder, and D. Becker, PRA 87, (2013). Summary

3 . Quantum control: generalities State-to-state ( state-selective ) control: How to steer a quantum system from a given initial- to a desired final state? Operator ( state-independent ) control: How to realize a pre-determined unitary transformation (target quantum gate)? p H(t) = H 0 + f j (t)h j j=1 f j (t) control fields The system is completely controllable if H(t) can give rise to an arbitrary unitary transformation on its Hilbert space H i.e., the reachable set R is equal to U(n) or SU(n) ( n = dim H )

4 . General controllability theorems U(t) = i[h 0 + p f j (t)h j ]U(t), U(0) = 1 (#) j=1 Lie algebra rank condition. Theorem. The reachable set R of a quantum control system described by Eq. (#) is the connected Lie group associated with the Lie algebra L 0 generated by. ih 0, ih 1,..., ih p, i.e., R = e L 0. complete (operator) controllability. Theorem. A system described by Eq. (#) is completely (operator) controllable iff L 0 = u(n) [ or L 0 = su(n) ], where L 0 is the Lie algebra generated. by ih 0, ih 1,..., ih p.

5 . Interacting qubit arrays Coupling between qubits: H int = i<j α,β J αβ ij σα i σβ j ( α, β = x, y, z ) qubit-qubit interaction Ising XY Heisenberg qubit system charge flux, charge-flux, phase, cavity spin, donor atom couplings beyond nearest neighbors can be induced using a quantum bus (e.g., cavity) [J. Majer et al., Nature (2007)]

6 . Local control in interacting systems: general aspects composite system S = C C with controls acting only on C total Hamiltonian: H = H S + p j=1 f C j (t)hc j S is completely controllable iff ih S and ih C j (j = 1,..., p) generate the Lie algebra L(S) of all skew-hermitian operators on S ih S, L(C) = L(S) L(C) = { ih C 1,..., ihc p } L A, B algebraic closure of the operator sets A and B

7 . Local control in qubit arrays with always-on interactions

8 . Complete controllability of XXZ Heisenberg chains H 0 = J N 1 i=1 (S i,x S i+1,x + S i,y S i+1,y + S i,z S i+1,z ) H c (t) = h x (t) S }{{}}{{} 1x f 1 (t) H 1 + h y (t) }{{} f 2 (t) S 1y }{{} H 2 H total(t) = H 0 + H c (t) Acting on the x- and y-components of an end spin in an XXZ Heisenberg spin chain renders the chain completely controllable! sufficient to show that the dimension of the dynamical Lie algebra L xy generated by { ih 0, is 1x, is 1y } is d 2 1 (d 2 N ) L xy = su(d) e L xy = SU(d) (complete controllability)

9 . Generalization to graphs Local controllability on a graph G = (S, E) by acting on C S p H = H S + fj C (t)hc j H S = j=1 n,m E H nm graph criterion of controllability (sufficient condition): algebraic property of H nm + topological property of G H S is algebraically propagating if for all n S and (n, m) E [ih nm, L(n)], L(n) = L(n, m) Heisenberg and Affleck-Kennedy-Lieb-Tasaki (AKLT) couplings are A.P.! S is controllable by acting on C if H S is A.P. and C is infecting D. Burgarth et. al., PRA 79, (R) (2009)

10 . Infecting and non-infecting subgraphs rules of infection:

11 . Control objectives (target gates) controlled-not on the last two qubits of the chain: ( ) CNOT N I... I }{{} N I X }{{} CNOT ( X σ x ) flip (NOT) of the last qubit X N requires only an x control! SWAP on the last two qubits: X SWAPN 1,N reminder: SWAP e i π 8 e i π (X X+Y Y +Z Z) 8 N = 3, = 1 case: dim L x = 18, basis { ih 0,..., ih 17 } Is there A L x such that SWAP2,3 = e A? 1XX + 1Y Y + 1ZZ = 1 2 (H 0 H 3 + H 6 H 16 + H 17 )

12 . Control pulses and fidelity maximization alternate x and y (or x only!) piecewise-constant controls: full time evolution (total time t f N t T ): U(t f ) = U y,nt /2U x,nt /2... U y,1 U x,1 [ ] U j,n e ihj,nt (j = x, y) gate fidelity: F (t f ) = 1 d tr [ U ] [ ] (t f )U target 0 F (t f ) 1 maximize F = F ({h x,n ; h y,n }) numerically frequency-filtered control fields: h j (t) = F 1[ f(ω)f[h j (t)] ] ideal low-pass filter: f(ω) = θ(ω + ω 0 ) θ(ω ω 0 )

13 . Toffoli-gate realization with superconducting qubits state of the art: two-qubit gates with F > 90% [ DiCarlo et al., (2009) ] TOFFOLI controlled-controlled-not a b 1 a b 1+ab trapped-ion [F 71%], photonic [F 81%] realizations in 2009! conventional 6 CNOTs + 10 single-qubit operations approach not feasible due to long gate times! Way out: use third level A. Fedorov et al., Nature 2012 : F = 64.5 ± 0.5 % M. D. Reed et al., Science 2012 : F = 78 ± 1 % Can quantum control be of help? F decoherence F exp( t g /T 2 )

14 . Three-qubit (transmon) circuit QED setup effective XY -type model: H 0 = i<j J ij (σ ix σ jx + σ iy σ jy ) H c (t) = 3 [ i=1 Ω (i) x J 12 = J 23 = J 30 MHz, J 13 5 MHz (t) σ ix + Ω (i) (t) σ iy y ] Ω 2 + x Ω2 130 MHz y VMS, A. Fedorov, A. Wallraff, and C. Bruder, PRB 85, (2012)

15 . Toffoli gate in circuit QED: results cutoff frequency: ω 0 = 500 MHz ω 0 17J! t g 140 ns F 99% t g = 75 ns F 92%

16 . Measurement-based quantum computation (MBQC) R. Raussendorf and H. J. Briegel, PRL 86, 5188 (2001) with local (single-qubit) measurements: MBQC one-way QC 2D cluster state is a universal resource for MBQC! Other candidates, e.g., the AKLT state, are difficult to produce in solid-state systems! H. J. Briegel et al., Nature Phys. 5, 19 (2009)

17 . Graph- and cluster states one-way quantum computing H. J. Briegel and R. Raussendorf, PRL 86, 910 (2001) initial preparation: G = {i,j} U (ij) PG + N + = ( )/ 2 U PG = diag(1, 1, 1, 1) correlation operators K i σ x i j nghd(i) σ z j satisfy K i ψ c = ± ψ c

18 . One-way QC: an illustration two-qubit cluster state: K 1 = σ x 1 σz 2, K 2 = σ z 1 σx 2 ψ c = 1 2 ( ) measure qubit 1 in a basis rotated by a SU(2) matrix: ( cos θ e U(θ, φ) iφ ) sin θ e iφ sin θ cos θ measurement results: m = 0 qubit 2 in ψ 0 = P (φ)r(θ) 0 m = 1 ψ 1 = σ1 zp ( φ)σx 1 R(θ) 1 P (φ) 1 ( ) 1 e iφ 2 1 e iφ ( ) cos θ sin θ R(θ) sin θ cos θ arbitrary single-qubit rotation on the second qubit can be performed by measuring the first one in an appropriate basis!

19 . Stabilizer Hamiltonian cluster states are eigenstates of the stabilizer Hamiltonian H stab = i K i How to generate a stabilizer Hamiltonian starting from a natural two-body spin-1/2 (qubit) Hamiltonian? H = H 0 + H int H 0 = i (Ω xσ x i + Ω yσ y i + ε iσ z i ) Ising: H int = J i XY : H int = J i Heisenberg: H int = J i σ z i σz i+1 (σ x i σx i+1 + σy i σy i+1 ) (σ x i σx i+1 + σy i σy i+1 + σz i σz i+1 )

20 . Stabilizer Hamiltonian from Ising-type interactions What e iθ i σz i σz i+1h s e iθ i σz i σz i+1 basic relations: amounts to? e iθσz 1 σz 2 σ x,y 1 eiθσz 1 σz 2 = cos(2θ)σ x,y 1 ± sin(2θ)σ y,x 1 σz 2 special case θ = π/4 increasing the order of the Pauli-matrix terms: e i π 4 σz 1 σz 2 σ x 1 e i π 4 σz 1 σz 2 = σ y 1 σz 2 ; e i π 4 σz 1 σz 2 σ y 1 ei π 4 σz 1 σz 2 = σ x 1 σ z 2 1D stabilizer Hamiltonian; 1D 2D straightforward! Q: But how can we physically generate ( induce effective dynamics of) H stab = e i(π/4) i σz i σz i+1h s e i(π/4) i σz i σz i+1?

21 . Stabilizer Hamiltonian as the effective Hamiltonian (a) H s H int H stab τ τ 1 τ 1 t Ising-interaction pulses with τ 1 π/(4j): H stab = e i π 4 state evolution: i σz i σz i+1h s e i π 4 i σz i σz i+1 ρ(0) τ 1H int τ H s τ 1 H int ρ(t = τ + 2τ1 ) pulse-induced effective evolution: ( e iτ H stab = exp i π ) ( σi z 4 σz i+1 e iτ Hs exp i π 4 i i σ z i σz i+1 ) unitary tsf. with a generator S, analytic operator function f(a): e S f(a) e S = f(e S A e S )

22 . Stabilizer Hamiltonian from XY -type interactions main difference from the Ising case: e iθ i (σx i σx i+1 +σy i σy i+1 ) does not factorize as [σi xσx i+1 + σy i σy i+1, σx i+1 σx i+2 + σy i+1 σy i+2 ] 0 H 2D step by step: stab 1. generate H stab for adjacent qubit pairs 2. connect the pairs to obtain H 1D stab 3. generate multiple H 1D and connect stab them pairwise into ladders 4. connect the ladders to obtain H 2D stab (b) σ z σ x (c) σ x σ z σ x σ z σ z the obtained H stab is twisted! the corresponding cluster state is twisted too! σ z σ z

23 . Case with always-on interactions Q: What if H 0 and H int cannot be switched on/off at will? A: The method can be applied in a stroboscopic fashion! based on the Baker-Campbell-Hausdorff (BCH) formula: e A e B = = exp ( A + B [A, B] [A, [A, B]] + 1 [B, [B, A]] +...) 12 realizing interaction pulse ( extracting H int ): A = i(h 0 + H int )τ, B = i( H 0 + H int )τ e A e B = ) exp (2iH int τ + (iτ ) 2 (iτ )3 [H 0, H int ] + 3 [H 0, [H 0, H int ]] +... repeating n-times with nωτ = π/4: in (e A e B ) ( ) n π k k-th term the effects of H 0 are eliminated! 4n

24 . Cluster-state fidelity: numerical results in the XY case F st (τ ) Ψ c U τ (δ) Ψ c 2 U τ (δ) e iτ H stab(π/4+δ) F st. N = 6 N = 8 N = σ analytical (perturbative) result: 1 F st δ 2 ( δ π/4 ) δ δ i (i = 1,..., N 1) averaged over random realizations of the δ i taken from a Gaussian distribution of width σ T. Tanamoto, D. Becker, VMS, and C. Bruder, PRA 86, (2012)

25 . Summary Local-control approach allows for an efficient realization of quantum gates in qubit arrays with XXZ Heisenberg interaction! 2D cluster states, a universal resource for MBQC, can be preserved with high fidelity in XY - and Ising-coupled qubit arrays! The two approaches to quantum control in interacting qubit arrays are complementary! V. M. Stojanović (Harvard) TI CMSA September 16, / 25