Exploring Quantum Chaos with Quantum Computers

Exploring Quantum Chaos with Quantum Computers Part II: Measuring signatures of quantum chaos on a quantum computer David Poulin Institute for Quantum Computing Perimeter Institute for Theoretical Physics MIT s Independent Activities Period, January 2004 p.1

Outline Classical chaos The quest for quantum signatures Readout problem DQC1 Random matrices & form factors Fidelity decay Quantum algorithm DQC1 algorithm Decoherence and quantum chaos Absence of entanglement Time permitting MIT s Independent Activities Period, January 2004 p.2

Classical chaos Lyapunov exponents. MIT s Independent Activities Period, January 2004 p.3

Classical chaos Lyapunov exponents. x+dx x dx(t) dx(t) dxe λt MIT s Independent Activities Period, January 2004 p.3

Classical chaos Lyapunov exponents. x+dx x dx(t) dx(t) dxe λt Kolmogorov-Sinai entropy, mixing of phase space. MIT s Independent Activities Period, January 2004 p.3

The quest for quantum signatures Problems with the classical signatures: MIT s Independent Activities Period, January 2004 p.4

The quest for quantum signatures Problems with the classical signatures: No notion of point in phase space, rather we have vectors in Hilbert space. MIT s Independent Activities Period, January 2004 p.4

The quest for quantum signatures Problems with the classical signatures: No notion of point in phase space, rather we have vectors in Hilbert space. Positivity of distributions over phase space is not preserved. MIT s Independent Activities Period, January 2004 p.4

The quest for quantum signatures Problems with the classical signatures: No notion of point in phase space, rather we have vectors in Hilbert space. Positivity of distributions over phase space is not preserved. Unitarity preserves distances between states: ψ(t) φ(t) = ψ(0) φ(0) MIT s Independent Activities Period, January 2004 p.4

Readout problem It has been known for some time that a quantum computer could be use to simulate the evolution of quantum systems (Zalka, Lloyd,...). Then what? U(t) = U n... U 2 U 1 }{{} Universal set MIT s Independent Activities Period, January 2004 p.5

Readout problem It has been known for some time that a quantum computer could be use to simulate the evolution of quantum systems (Zalka, Lloyd,...). U(t) = U n... U 2 U 1 }{{} Universal set Then what? Classical simulation: ψ = (0.2677 + 0.002i,... 0.00098538 0.1i) T. MIT s Independent Activities Period, January 2004 p.5

Readout problem It has been known for some time that a quantum computer could be use to simulate the evolution of quantum systems (Zalka, Lloyd,...). U(t) = U n... U 2 U 1 }{{} Universal set Then what? Classical simulation: ψ = (0.2677 + 0.002i,... 0.00098538 0.1i) T. Quantum lab: single or multiple (macroscopically many) systems described by ψ. MIT s Independent Activities Period, January 2004 p.5

Readout problem It has been known for some time that a quantum computer could be use to simulate the evolution of quantum systems (Zalka, Lloyd,...). U(t) = U n... U 2 U 1 }{{} Universal set Then what? Classical simulation: ψ = (0.2677 + 0.002i,... 0.00098538 0.1i) T. Quantum lab: single or multiple (macroscopically many) systems described by ψ. Quantum computer: Ability to simulate evolution. MIT s Independent Activities Period, January 2004 p.5

Readout problem Do we need to prepare a special initial state? Low temperature transport properties ψ 0 e i(h 0+V ) ψ k. MIT s Independent Activities Period, January 2004 p.6

Readout problem 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 accumulate statistics? How many? Spectrum of a molecule. etc. The ability to efficiently simulate the dynamics U(t) alone can be used to extract interesting physical quantities. MIT s Independent Activities Period, January 2004 p.6

DQC1 Liquid state NMR QIP Thermal state ρ = 1 Z e βh. MIT s Independent Activities Period, January 2004 p.7

DQC1 Liquid state NMR QIP Thermal state ρ = 1 Z e βh. H µ i i B + J i j ij µ i µ j and β 1, so ( ρ 1 1l β ) ω i σ zi Z i MIT s Independent Activities Period, January 2004 p.7

DQC1 Liquid state NMR QIP Thermal state ρ = 1 Z e βh. H µ i i B + J i j ij µ i µ j and β 1, so ( ρ 1 1l β ) ω i σ zi Z i Using a gradient field, ρ 1 Z ( 1l + βωn ) 0... 00 0... 00 2n D.G. Cory, A.F. Fahmy, and T.F. Havel, 1997, & N.A. Gershenfeld and I.L. Chuang 1997 MIT s Independent Activities Period, January 2004 p.7

DQC1 This exponential decay can be avoided using algorithmic cooling, but the overhead is 10 10. MIT s Independent Activities Period, January 2004 p.8

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 MIT s Independent Activities Period, January 2004 p.8

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 1l + ɛ 0 0 2 1 2 n1l MIT s Independent Activities Period, January 2004 p.8

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 1l + ɛ 0 0 2 1 2 n1l 2. No projective measurement, rather σ z within finite accuracy µ. MIT s Independent Activities Period, January 2004 p.8

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 1l + ɛ 0 0 2 1 2 n1l 2. No projective measurement, rather σ z within finite accuracy µ. Can we do something non trivial with this? MIT s Independent Activities Period, January 2004 p.8

DQC1 Remarks: E. Knill and R. Laflamme, 1998 1. Not robust against noise. MIT s Independent Activities Period, January 2004 p.9

DQC1 Remarks: E. Knill and R. Laflamme, 1998 1. Not robust against noise. 2. Measurement accuracy µ δ: cost (µ/δ) 2. MIT s Independent Activities Period, January 2004 p.9

DQC1 Remarks: E. Knill and R. Laflamme, 1998 1. Not robust against noise. 2. Measurement accuracy µ δ: cost (µ/δ) 2. 3. Pseudo-pure state pure state ρ = 0 0 1 2 n 1l: cost poly(n). MIT s Independent Activities Period, January 2004 p.9

DQC1 Remarks: E. Knill and R. Laflamme, 1998 1. Not robust against noise. 2. Measurement accuracy µ δ: cost (µ/δ) 2. 3. Pseudo-pure state pure state ρ = 0 0 1 2 n 1l: cost poly(n). 4. One pure qubit log(n) pure qubits: cost poly(n). MIT s Independent Activities Period, January 2004 p.9

Trace circuit 0 0 H H σ k ρ U C.Miquel, J.P.Paz, M.Saraceno, E. Knill, R. Laflamme, and C.Negrevergne, 2002 MIT s Independent Activities Period, January 2004 p.10

Trace circuit 0 0 H H σ k ρ U C.Miquel, J.P.Paz, M.Saraceno, E. Knill, R. Laflamme, and C.Negrevergne, 2002 ρ 0 = 0 0 ρ 1 ( 0 0 ρ + 0 1 ρ + 1 0 ρ + 1 1 ρ) 2 1 2 ( 0 0 ρ + 0 1 ρu + 1 0 Uρ + 1 1 UρU ) = ρ f MIT s Independent Activities Period, January 2004 p.10

Trace circuit 0 0 H H σ k ρ U C.Miquel, J.P.Paz, M.Saraceno, E. Knill, R. Laflamme, and C.Negrevergne, 2002 ρ 0 = 0 0 ρ 1 ( 0 0 ρ + 0 1 ρ + 1 0 ρ + 1 1 ρ) 2 1 2 ( 0 0 ρ + 0 1 ρu + 1 0 Uρ + 1 1 UρU ) = ρ f ρ 1 = T r n {ρ f } = 1 2 ( 0 0 + 0 1 T r{ρu } + 1 0 T r{uρ} + 1 1 ) MIT s Independent Activities Period, January 2004 p.10

Trace circuit 0 0 ρ With γ = T r{uρ}, H H σ k U MIT s Independent Activities Period, January 2004 p.11

Trace circuit 0 0 ρ With γ = T r{uρ}, H H σ k U ρ 1 = 1 2 ( 0 0 + 0 1 γ + 1 0 γ + 1 1 ) MIT s Independent Activities Period, January 2004 p.11

Trace circuit 0 0 ρ With γ = T r{uρ}, H H σ k U ρ 1 = 1 2 ( 0 0 + 0 1 γ + 1 0 γ + 1 1 ) ( ) = 1 1 γ 2 γ 1 MIT s Independent Activities Period, January 2004 p.11

Trace circuit 0 0 ρ With γ = T r{uρ}, H H σ k U ρ 1 = 1 2 ( 0 0 + 0 1 γ + 1 0 γ + 1 1 ) ( ) = 1 1 γ 2 γ 1 ( ) 1 1 + Re{γ} iim{γ} 2 iim{γ} 1 Re{γ} MIT s Independent Activities Period, January 2004 p.11

Trace circuit 0 0 H H σ k ρ ρ 1 = 1 2 ( U 1 + Re{γ} iim{γ} iim{γ} 1 Re{γ} ) MIT s Independent Activities Period, January 2004 p.12

Trace circuit 0 0 H H σ k ρ ρ 1 = 1 2 ( U 1 + Re{γ} iim{γ} iim{γ} 1 Re{γ} ) k = z : σ z = T r{ρ 1σ z } = Re{γ} = Re [T r{ρu}] k = y : σ y = T r{ρ 1σ y } = Im{γ} = Im [T r{ρu}] MIT s Independent Activities Period, January 2004 p.12

Trace circuit 0 0 H H σ k ρ ρ 1 = 1 2 ( U 1 + Re{γ} iim{γ} iim{γ} 1 Re{γ} ) k = z : σ z = T r{ρ 1σ z } = Re{γ} = Re [T r{ρu}] k = y : 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. MIT s Independent Activities Period, January 2004 p.12

Trace circuit T r{ρu} MIT s Independent Activities Period, January 2004 p.13

Trace circuit T r{ρu} Control U and learn about ρ: quantum state tomography. MIT s Independent Activities Period, January 2004 p.13

Trace circuit T r{ρu} Control U and learn about ρ: quantum state tomography. Control ρ and learn about U: scattering or quantum process tomography. MIT s Independent Activities Period, January 2004 p.13

Trace circuit T r{ρu} Control U and learn about ρ: quantum state tomography. Control ρ and learn about U: scattering or quantum process tomography. Can we get useful partial information about U? (Tomography is exponentially hard.) MIT s Independent Activities Period, January 2004 p.13

Trace circuit Let s see what we can do in the DQC1 model. 0 0 1 1l N H H σ k U MIT s Independent Activities Period, January 2004 p.14

Trace circuit Let s see what we can do in the DQC1 model. 0 0 H H σ k 1 1l U N 1 N T r{u} = 1 N N j=1 λ j MIT s Independent Activities Period, January 2004 p.14

Quantum control QIP is about manipulating complex quantum systems. MIT s Independent Activities Period, January 2004 p.15

Quantum control QIP is about manipulating complex quantum systems. Control a quantum system = exploit its symmetries (QEC, NSS,... ) MIT s Independent Activities Period, January 2004 p.15

Quantum control QIP is about manipulating complex quantum systems. Control a quantum system = exploit its symmetries (QEC, NSS,... ) Does my system possess any symmetries? MIT s Independent Activities Period, January 2004 p.15

Quantum control QIP is about manipulating complex quantum systems. Control a quantum system = exploit its symmetries (QEC, NSS,... ) Does my system possess any symmetries? Systems with as many symmetries as degrees of freedom are integrable. MIT s Independent Activities Period, January 2004 p.15

Quantum control QIP is about manipulating complex quantum systems. Control a quantum system = exploit its symmetries (QEC, NSS,... ) Does my system possess any symmetries? Systems with as many symmetries as degrees of freedom are integrable. Chaotic systems possess no or very few symmetries: They are hard to control. MIT s Independent Activities Period, January 2004 p.15

Looking for symmetries Resonances of Thorium 232 Garg et al., Phys. Rev. 134, B85, 1964. MIT s Independent Activities Period, January 2004 p.16

Looking for symmetries List of resonances Level spacing MIT s Independent Activities Period, January 2004 p.17

Looking for symmetries Wigner s idea: The main characteristics of the spectrum can be reproduced by random matrices which possess the same symmetries of the system. MIT s Independent Activities Period, January 2004 p.18

Looking for symmetries Wigner s idea: The main characteristics of the spectrum can be reproduced by random matrices which possess the same symmetries of the system. No symmetry = Level repulsion U d P r(d) d MIT s Independent Activities Period, January 2004 p.18

Looking for symmetries Symmetries = Block diagonal { } iht U = exp = h U 1 U 2... U d MIT s Independent Activities Period, January 2004 p.19

Looking for symmetries Symmetries = Block diagonal { } iht U = exp = h U 1 U 2... U d U 1 + U 2 +... = U d P r(d) d MIT s Independent Activities Period, January 2004 p.19

Looking for symmetries 1 N T r{u} = 1 N = j v j j eiφ j MIT s Independent Activities Period, January 2004 p.20

Looking for symmetries 1 N T r{u} = 1 N = j v j j eiφ j 5 0 5 10 15 20 25 30 30 20 10 0 10 MIT s Independent Activities Period, January 2004 p.20

Looking for symmetries 1 N T r{u} = 1 N = j v j j eiφ j With symmetries (regular): 5 0 5 10 15 20 25 30 30 20 10 0 10 1 N T r{u} = 1 N N = 1 N MIT s Independent Activities Period, January 2004 p.20

Looking for symmetries 1 N T r{u} = 1 N = j v j j eiφ j With symmetries (regular): No symmetries (chaotic): 5 0 5 10 15 20 25 30 30 20 10 0 10 1 N T r{u} = 1 N N = 1 N 1 N T r{u} = 1 N O(1) = 1 N MIT s Independent Activities Period, January 2004 p.20

Looking for symmetries 1 N T r{u} = 1 N = j v j j eiφ j With symmetries (regular): No symmetries (chaotic): 5 0 5 10 15 20 25 30 30 20 10 0 10 1 N T r{u} = 1 N Requires accuracy 1 N so overhead is N. N = 1 N 1 N T r{u} = 1 N O(1) = 1 N D. Poulin, R.Laflamme, G.J. Milburn, and J.P. Paz, PRA 68, 22302 (2003). MIT s Independent Activities Period, January 2004 p.20

Fidelity decay Quantum map U, e.g. U = exp{ iht} MIT s Independent Activities Period, January 2004 p.21

Fidelity decay Quantum map U, e.g. U = exp{ iht} Perturbed quantum map U p, e.g. U p = U exp{ iδv } }{{} K MIT s Independent Activities Period, January 2004 p.21

Fidelity decay Quantum map U, e.g. U = exp{ iht} Perturbed quantum map U p, e.g. U p = U exp{ iδv } }{{} K F n (ψ) = ψ (U n ) U n p ψ 2 MIT s Independent Activities Period, January 2004 p.21

Fidelity decay Quantum map U, e.g. U = exp{ iht} Perturbed quantum map U p, e.g. U p = U exp{ iδv } }{{} K F n (ψ) = ψ (U n ) U n p ψ 2 F n = F n (ψ)dψ = T r{(u n ) U n p } 2 + N N 2 + N MIT s Independent Activities Period, January 2004 p.21

Fidelity decay Peres conjecture: F n (ψ) is a signature of quantum chaos (dynamical instability). MIT s Independent Activities Period, January 2004 p.22

Fidelity decay Peres conjecture: F n (ψ) is a signature of quantum chaos (dynamical instability). Chaotic systems have an exponential decay exp( Γn). MIT s Independent Activities Period, January 2004 p.22

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. J. Emerson, Y.S. Weinstein, S. Lloyd, and D.G. Cory, 2002 MIT s Independent Activities Period, January 2004 p.22

Trace circuit D. Poulin, R. Blume-Kohout, R. Laflamme, and H. Ollivier, 2003. 0 0 ρ H H σ k U MIT s Independent Activities Period, January 2004 p.23

Trace circuit D. Poulin, R. Blume-Kohout, R. Laflamme, and H. Ollivier, 2003. 0 0 ρ H H σ k U k = y : σ y = Im [T r{ρu}] k = z : σ z = Re [T r{ρu}] MIT s Independent Activities Period, January 2004 p.23

Trace circuit D. Poulin, R. Blume-Kohout, R. Laflamme, and H. Ollivier, 2003. 0 0 ρ H H σ k U k = y : σ y = Im [T r{ρu}] k = z : σ z = Re [T r{ρu}] F n = T r{(u n ) U n p } 2 + N N 2 + N, U p = UK MIT s Independent Activities Period, January 2004 p.23

Quantum circuit 0 H σ k 1l N U K U K... U K U U... U } {{ } n } {{ } n MIT s Independent Activities Period, January 2004 p.24

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. MIT s Independent Activities Period, January 2004 p.24

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. MIT s Independent Activities Period, January 2004 p.24

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 (n) ɛ 2 MIT s Independent Activities Period, January 2004 p.24

Decoherence Decoherence is the phenomenon responsible for the destruction of coherent quantum superpositions in favor of classical mixtures. MIT s Independent Activities Period, January 2004 p.25

Decoherence Decoherence is the phenomenon responsible for the destruction of coherent quantum superpositions in favor of classical mixtures. It arises from a coupling between the system and its environment (uncontrolled degrees of freedom. The stronger the coupling to the environment, the faster decoherence takes place. MIT s Independent Activities Period, January 2004 p.25

Decoherence Decoherence is the phenomenon responsible for the destruction of coherent quantum superpositions in favor of classical mixtures. It arises from a coupling between the system and its environment (uncontrolled degrees of freedom. The stronger the coupling to the environment, the faster decoherence takes place. α 0 + β 1 0 S E α 00 + β 11 ρ S α 2 0 0 + β 2 1 1 MIT s Independent Activities Period, January 2004 p.25

Decoherence ρ 0 ρ n 1l N U K... U K U K MIT s Independent Activities Period, January 2004 p.26

Decoherence 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 ) MIT s Independent Activities Period, January 2004 p.26

Decoherence 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! MIT s Independent Activities Period, January 2004 p.26

Entanglement α 0 + β 1 1l N U K... 1 1 U 2 K 2 U k K k MIT s Independent Activities Period, January 2004 p.27

Entanglement α 0 + β 1 1l N U K... 1 1 U 2 K 2 U k K k The final state is ρ k = 1 N j α j α j φ j φ j where α j = α 0 + βe is j 1 and the eis j are the eigenvalues of S = U k P k... U 2 P 2 U 1 P 1 U 1U 2... U k. MIT s Independent Activities Period, January 2004 p.27

Entanglement There is no entanglement between the control qubit and the rest of the computer. MIT s Independent Activities Period, January 2004 p.28

Entanglement There is no entanglement between the control qubit and the rest of the computer. 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.) MIT s Independent Activities Period, January 2004 p.28

Entanglement There is no entanglement between the control qubit and the rest of the computer. 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. MIT s Independent Activities Period, January 2004 p.28

Entanglement There is no entanglement between the control qubit and the rest of the computer. 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:? MIT s Independent Activities Period, January 2004 p.28

Entanglement There is no entanglement between the control qubit and the rest of the computer. 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. MIT s Independent Activities Period, January 2004 p.28

Quantum probe ρ 0 1l N U K... U K U K MIT s Independent Activities Period, January 2004 p.29

Quantum probe ρ 0 1l N U K... U K U K H = J 1j Z 1 Z j j>1 }{{} perturbation + Unknown U {}}{ H(i, j 1) MIT s Independent Activities Period, January 2004 p.29