Exploring Quantum Control with Quantum Information Processors

Exploring Quantum Control with Quantum Information Processors David Poulin Institute for Quantum Computing Perimeter Institute for Theoretical Physics IBM, March 2004 p.1

Two aspects of quantum information Practical Build a quantum computer. Design new quantum algorithms. Invent useful quantum protocols. Design quantum error corrections techniques. IBM, March 2004 p.2

Two aspects of quantum information Practical Build a quantum computer. Design new quantum algorithms. Invent useful quantum protocols. Design quantum error corrections techniques. Fundamental Use the byproducts to study fundamental physics. Foundation of quantum mechanics: what gives quantum mechanics extra computational power? Explain renormalization groups as error correction. IBM, March 2004 p.2

Concrete example Design useful task for a present days quantum information processors: Can we do something useful in DQC1? Is entanglement responsible for quantum computational speed up? Set milestones for the construction of a quantum computer. How can we improve our control over quantum systems? (QECC, NSS, bang-bang) New tools to understand the role of symmetries of quantum systems. Relation between dynamical stability and symmetries. IBM, March 2004 p.3

Outline Simulating quantum systems: the readout problem. The QDC1 model of computation. The elementary scattering circuit. Looking for symmetries in complex quantum systems. Random matrix conjecture. Estimating the form factors. Hypersensitivity to perturbations: fidelity decay. DQC1 algorithm. Quantum information processors as quantum probes. Decoherence and dynamical instability. Entanglement, decoherence and quantum-computational speed-up. Conclusion. IBM, March 2004 p.4

Readout problem In a physical experiment, a system evolves from its initial state and is finally measured in some basis. IBM, March 2004 p.5

Readout problem In a physical experiment, a system evolves from its initial state and is finally measured in some basis. We know that quantum computer can be use to simulate the evolution of some quantum systems (Zalka, Lloyd,...). U(t) = U n... U 2 U 1 }{{} Universal set Do we need to prepare a special initial state? Low temperature transport properties ψ 0 e i(h 0+V ) ψ k. Do we need to know the spectral projectors of the measured quantity? IBM, March 2004 p.5

DQC1 Liquid state NMR QIP The initial state is thermal ρ = 1 Z e βh. IBM, March 2004 p.6

DQC1 Liquid state NMR QIP The initial state is thermal ρ = 1 Z e βh. H i µ i B + i j J ij µ i µ j, ω J and β 1, so ( ρ 1 1l β ) ω i σ zi Z i IBM, March 2004 p.6

DQC1 Liquid state NMR QIP The initial state is thermal ρ = 1 Z e βh. H i µ i B + i j J ij µ i µ j, ω J and β 1, so ( ρ 1 1l β ) ω i σ zi Z i Apply an inhomogeneous magnetic field to induce a random phase: ρ 1 ( 1l + βωn ) 0... 00 0... 00 Z 2n Pseudo pure state, the identity part does not contribute to the computation. D.G. Cory, A.F. Fahmy, and T.F. Havel, 1997, & N.A. Gershenfeld and I.L. Chuang, 1997. IBM, March 2004 p.6

DQC1 This exponential decay can be avoided using algorithmic cooling, but the overhead is 10 10. IBM, March 2004 p.7

DQC1 This exponential decay can be avoided using algorithmic cooling, but the overhead is 10 10. Add restrictions to the standard model of quantum computation IBM, March 2004 p.7

DQC1 This exponential decay can be avoided using algorithmic cooling, but the overhead is 10 10. Add restrictions to the standard model of quantum computation 1. Only a single pseudo-pure qubit. ρ = ( 1 ɛ 2 ) 1l + ɛ 0 0 1 2 n 1l 2. No projective measurement, rather σ z within finite accuracy µ. IBM, March 2004 p.7

DQC1 Remarks: 1. Not robust against noise. Nowhere to dump the entropy! IBM, March 2004 p.8

DQC1 Remarks: 1. Not robust against noise. Nowhere to dump the entropy! 2. Measurement accuracy µ δ: cost (µ/δ) 2. IBM, March 2004 p.8

DQC1 Remarks: 1. Not robust against noise. Nowhere to dump the entropy! 2. Measurement accuracy µ δ: cost (µ/δ) 2. 3. Pseudo-pure state pure state ρ = 0 0 1 2 n 1l: cost 1/ɛ 2. IBM, March 2004 p.8

DQC1 Remarks: 1. Not robust against noise. Nowhere to dump the entropy! 2. Measurement accuracy µ δ: cost (µ/δ) 2. 3. Pseudo-pure state pure state ρ = 0 0 1 2 n 1l: cost 1/ɛ 2. 4. One pure qubit log(n) pure qubits: cost poly(n). From a computational complexity point of view: 00... 0 00... 0 1 }{{} 2 n 1l ln n IBM, March 2004 p.8

Scattering circuit 0 0 ρ H H σ k U γ = T r{uρ} IBM, March 2004 p.9

Scattering circuit 0 0 ρ H H σ k U γ = T r{uρ} ( 1 + Re{γ} iim{γ} ) ρ 1 = 1 2 iim{γ} 1 Re{γ} IBM, March 2004 p.9

Scattering circuit 0 0 ρ H H σ k U γ = T r{uρ} ( 1 + Re{γ} iim{γ} ) ρ 1 = 1 2 iim{γ} 1 Re{γ} k = z k = y : σ z = T r{ρ 1σ z } = Re{γ} = Re [T r{ρu}] : σ y = T r{ρ 1σ y } = Im{γ} = Im [T r{ρu}] IBM, March 2004 p.9

Scattering circuit 0 0 ρ H H σ k U γ = T r{uρ} ( 1 + Re{γ} iim{γ} ) ρ 1 = 1 2 iim{γ} 1 Re{γ} k = z k = y : σ z = T r{ρ 1σ z } = Re{γ} = Re [T r{ρu}] : σ y = T r{ρ 1σ y } = Im{γ} = Im [T r{ρu}] If U can be implemented efficiently and ρ prepared efficiently, we can evaluate T r{ρu} within accuracy δ in time poly(n)/δ 2. IBM, March 2004 p.9

Scattering circuit We can evaluate T r{ρu} IBM, March 2004 p.10

Scattering circuit We can evaluate T r{ρu} Control U and learn about ρ: quantum state tomography. IBM, March 2004 p.10

Scattering circuit We can evaluate T r{ρu} Control U and learn about ρ: quantum state tomography. Control ρ and learn about U: quantum process tomography. C.Miquel, J.P.Paz, M.Saraceno, E. Knill, R. Laflamme, and C.Negrevergne, 2002 These algorithms require an exponential repetition of the scattering circuit. IBM, March 2004 p.10

Quantum control Quantum information processing is the art of manipulating complex quantum systems. IBM, March 2004 p.11

Quantum control Quantum information processing is the art of manipulating complex quantum systems. Control a quantum system = exploit its symmetries (QECC, NSS, bang-bang) IBM, March 2004 p.11

Quantum control Quantum information processing is the art of manipulating complex quantum systems. Control a quantum system = exploit its symmetries (QECC, NSS, bang-bang) Does my system possess any symmetries? Systems with as many symmetries as degrees of freedom are integrable. These are good candidates for quantum information processors. IBM, March 2004 p.11

Looking for symmetries Resonances of Thorium 232 Garg et al., Phys. Rev. 134, B85, 1964. IBM, March 2004 p.12

Looking for symmetries List of resonances Level spacing IBM, March 2004 p.13

Looking for symmetries Wigner s intuition: The main characteristics of the spectrum of can be reproduced by random matrices which possess the same symmetries as the system. IBM, March 2004 p.14

Looking for symmetries Wigner s intuition: The main characteristics of the spectrum of can be reproduced by random matrices which possess the same symmetries as the system. f(u) f(v )P (V )dv. P (V ) 0 only for those V with the same symmetries as U. P (V ) is determined with the maximum entropy principle given the above constraint. IBM, March 2004 p.14

Looking for symmetries Symmetries = Block diagonal { } iht U = exp h = U 1 U 2... U d IBM, March 2004 p.15

Looking for symmetries Symmetries = Block diagonal { } iht U = exp h = U 1 U 2... U d d U 1 + U 2 +... = U P r(d) Mixture of i.i.d variables Poisson distribution. d IBM, March 2004 p.15

Looking for symmetries 1 N T r{u} = 1 N = 1 N j eiφ j j v j IBM, March 2004 p.16

Looking for symmetries 1 N T r{u} = 1 N = 1 N j eiφ j j v j 5 0 5 10 15 20 25 30 30 20 10 0 10 IBM, March 2004 p.16

Looking for symmetries 1 N T r{u} = 1 N = 1 N j eiφ j j v j 5 0 5 10 15 20 25 30 30 20 10 0 10 1 With symmetries (regular): N T r{u} = 1 N N = 1 N IBM, March 2004 p.16

Looking for symmetries 1 N T r{u} = 1 N = 1 N j eiφ j j v j 5 0 5 10 15 20 25 30 30 20 10 0 10 1 With symmetries (regular): N T r{u} = 1 N N = 1 N No symmetries (chaotic): 1 N T r{u} = 1 N O(1) = 1 N IBM, March 2004 p.16

Looking for symmetries 1 N T r{u} = 1 N = 1 N j eiφ j j v j 5 0 5 10 15 20 25 30 30 20 10 0 10 1 With symmetries (regular): N T r{u} = 1 N N = 1 N No symmetries (chaotic): 1 N T r{u} = 1 N O(1) = 1 N Requires accuracy 1 N so overhead is N: square-root improvement over brute force classical computation. D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003). IBM, March 2004 p.16

Fidelity decay Quantum map U, e.g. U = exp{ iht}. How sensitive is U to perturbation? IBM, March 2004 p.17

Fidelity decay Quantum map U, e.g. U = exp{ iht}. How sensitive is U to perturbation? Perturbed quantum map U p, e.g. U p = U exp{ iδv } }{{} K IBM, March 2004 p.17

Fidelity decay Quantum map U, e.g. U = exp{ iht}. How sensitive is U to perturbation? Perturbed quantum map U p, e.g. U p = U exp{ iδv } }{{} K Fidelity after n steps (Lodschmidt echo): F n (ψ) = ψ (U n ) U n p ψ 2 IBM, March 2004 p.17

Fidelity decay Peres conjecture: F n (ψ) is a signature of quantum chaos (dynamical instability). Chaotic systems have an exponential decay exp( Γn). Regular systems have a polynomial decay 1/poly(n). Fidelity decay measures the relative randomness of U and K. Universal response to perturbation. Simple perturbation produce good signature of quantum chaos. IBM, March 2004 p.18

Trace circuit D. Poulin, R. Blume-Kohout, R. Laflamme, and H. Ollivier, 2003. 0 0 1l 2 n H H σ k U k = y : σ y = 1 N Im [T r{u}] k = z : σ z = 1 Re [T r{u}] N T r{(u n ) Up n } 2 + N F n = N 2, U p = UK + N IBM, March 2004 p.19

Trace circuit D. Poulin, R. Blume-Kohout, R. Laflamme, and H. Ollivier, 2003. 0 0 1l 2 n H H σ k U k = y : σ y = 1 N Im [T r{u}] k = z : σ z = 1 Re [T r{u}] N T r{(u n ) Up n } 2 + N F n = N 2, U p = UK + N All we have to do it replace U in the above circuit by (U n ) (UK) n. IBM, March 2004 p.19

Quantum circuit 0 H σ k 1l N U K U K... U K U U... U } {{ } n } {{ } n IBM, March 2004 p.20

Quantum circuit 0 1l N H U K U K... U K U U... U } {{ } n } {{ } n σ k Since the U s and U s annihilate each others when the K s are absent. IBM, March 2004 p.20

Quantum circuit 0 1l N H U K U K... U K } {{ } n σ k Since the U s and U s annihilate each others when the K s are absent. Since the U are made after the final measurement. IBM, March 2004 p.20

Quantum circuit 0 1l N H U K U K... U K } {{ } n σ k Since the U s and U s annihilate each others when the K s are absent. Since the U are made after the final measurement. Measure F n within accuracy ɛ with error probability p in time ( ) log(1/p) no T ɛ 2 IBM, March 2004 p.20

Quantum probe ρ 0 1l N U K U K... U K IBM, March 2004 p.21

Quantum probe ρ 0 1l N U K U K... U K Use a QIP to simulate a system with Hamiltonian H. IBM, March 2004 p.21

Quantum probe ρ 0 1l N U K U K... U K Use a QIP to simulate a system with Hamiltonian H. Use a quantum probe to investigate the dynamical stability of a system whose Hamiltonian is unknown. H = j>1 J 1j Z 1 Z j + Unknown U {}}{ H(i, j 1) }{{} perturbation IBM, March 2004 p.21

Decoherence We use the circuit to model a system interacting with an environment, the system is initially in a pure state α 0 + β 1 : ρ 0 ρ n 1l N U K U K... U K IBM, March 2004 p.22

Decoherence We use the circuit to model a system interacting with an environment, the system is initially in a pure state α 0 + β 1 : ρ 0 ρ n 1l N U K U K... U K ρ 0 = ( α 2 αβ α β β 2 ) ρ n = ( α 2 αβ γ(n) α βγ(n) β 2 ) γ(n) = { F n } 1/2: The decoherence rate is governed by the average fidelity decay rate of the environment! IBM, March 2004 p.22

Entanglement α 0 + β 1 1l N U 1 K 1 U 2 K 2... U k K k IBM, March 2004 p.23

Entanglement α 0 + β 1 1l N U 1 K 1 U 2 K 2... U k K k Using the "phase kick-back", we see that the final state is ρ k = 1 N α j α j φ j φ j j where α j = α 0 + βe is j 1 and the e is j S = U k P k... U 2 P 2 U 1 P 1 U 1 U 2... U k. are the eigenvalues of IBM, March 2004 p.23

Entanglement Decoherence does not require entanglement between the system and its environment: classical correlations are enough. IBM, March 2004 p.24

Entanglement Decoherence does not require entanglement between the system and its environment: classical correlations are enough. Is entanglement necessary for quantum computational speed-up? (If speed-up there is.) Pure states: YES. Mixed states:? This is an argument in favor of no... but not a proof. Is there entanglement between other parts of the information processor? (Depends on the system of interest, i.e. U.) IBM, March 2004 p.24

Conclusion DQC1 is interesting. Measure the form factors to learn about symmetries. Measure fidelity decay to learn about dynamical stability. Simple quantum information processors can help to perform experiments. Decoherence rate is influenced by the dynamical properties of the environment. Decoherence does not require entenglement. Entenglement is not the key ingredient to quantum computational speed-up. We can learn about fundamental physics by working on quantum computation, and vice versa. IBM, March 2004 p.25