One-component Quantum Mechanics Universal Leakage Elimination Operator

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I introduce universal leakage elimination operation for all linear system, including quantum open systems. Part of the talk has not yet published.

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ONE COMPONENT or 1-d QUANTUM MECHANICS AND A UNIVERSAL LEAKAGE ELIMINATION OPERATOR Lian-Ao Wu Department of Theoretical Physics The Basque Country University EHU/UPV Bilbao Spain Ikerbasque Basque Foundation for Science

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BILBAO THE BASQUE COUNTRY “ONE-DIMENSIONAL” CITY

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OUTLINE  Feshbach P-Q partitioning technique and One- Component Exact Dynamical Equation different perspective to look into finite dimensional QM  Introduce Leakage Elimination Operator LEO  General condition for a type of quantum control: Keeping a quantum system walking on a given state |At in the presence of environment. Examples:   Fast pulses and noise preserved quantum memory   Noise or fast pulses induced Adiabaticity   Application to the 3SAT algorithm  Summary

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FESHBACH P-Q PARTITIONING Consider a linear equation of motion X P Q ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ M h 0 | R W | D ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ h 0 1×1 R 1× N −1... MX ∂ X ∂t N - dimensional We can write down the solution matrix as We can also set otherwise rotate it to h 0 0 M 0 | R W | D− h 0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

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LEAKAGE ELIMINATION OPERATOR LEO Consider a Hamiltonian L.-A. Wu et al. PRL 89 127901 H LEO diag ht0...0 If ht∝ δ t− nτ ∑ the propagator is an LEO R L −1 0 0 I ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ R L RW R L −RW

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SUCH THAT ideally R L e − Mτ R L e − Mτ τ ⎯ → ⎯ 0 ⎯ → ⎯⎯ exp−2τ 0 0 0 D ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Leakage from P space to Q is eliminated. Examples: Qubit H LEO ht 1 1 or σ z Approximate LEO H LEO diag hth 1 t...h N t If |ht |≫ |h i t |

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ONE-COMPONENT DYNAMICAL EQUATION: P01Q00 ∂Pt ∂t htPt+ d 0 t ∫ sgtsPs where gts RtGtsW sht from LEO Gts T ← exp D s t ∫ ′ s d ′ s . OR ∂pt ∂t d 0 t ∫ sgtsps where gts e h s t ∫ ′ s d ′ s gts and pt exp h 0 t ∫ ′ s d ′ s Pt.

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DERIVATION OF THE EQUATION PhP+RQ QWP+DQ Qt GtsW sPsds 0 t ∫ QGttW tPt+DtQt where GtsDtGts and Gtt 1 MX ∂ X ∂t

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Classical Harmonic oscillator Schrodinger’s equation Liouvilles equation for open quantum system Quantum state diffusion equation for open system Generalized coordinates and momenta H: Hamiltonian wave function. L: super-operator Liouvillian. density matrix of the system. h: non-Hermitan effective hamiltonian z is a state of bath M −iH M L M −ih eff X→ψ X→ρ X→ zψ Examples of linear equations of motion M 0 1/m −ω 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ X → η ξ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥

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REFERENCES   L. -A. Wu et al. Master Equation and Control an Open System with Leakage Phys. Rev. Lett. 102 080405 2009.   J. Jing and L. -A. Wu Control of decoherence with no control Sci. Rep. 3 2746 2013.   J.Jing A. Bishop and L.-A. Wu Nonperturbative dynamical decoupling with random control Sci. Rep. 46299 2014.   J. Jing L.-A. Wu T. Yu J. Q. You Z.-M. Wang L. Garcia One- component dynamical equation and noise-induced adiabaticity Phys. Rev. A 89 032110 2014.  H. F. Wang and L.-A. Wu Fast quantum algorithm for EC3 problem with trapped ions. arXiv:1412.1722  J. Jing L.-A. Wu M. Byrd J. Q. You T. Yu Z.-M. Wang Nonperturbative Leakage Elimination Operators and Control of a Three-Level System Phys. Rev. Lett. 114 190502 2015.

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ONE-COMPONENT EQUATION. tracing footprint of a target state In quantum cases we can track a target time-evolving state At closed system or QSD At At general master equation The wave function evolves ψ C A At + ..... or ρ C A At At + ... At A ⊥ t to rewrite MX ∂ t X We set p C A ∂ t C A t d 0 t ∫ sg tsC A s where g ts gtse h s t ∫ ′ s d ′ s E1: If g ts −k∂ t C A t −k C A sds 0 t ∫ Let x C A sds 0 t ∫ x −kx E2 : g ts −λe hsds s t ∫ δ t− sideal Markovian ∂ t C A t −λC A t

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GENERAL CONDITION FOR FULL CONTROL d 0 t ∫ se − h 0 s ∫ ′ s d ′ s gtsC A s 0 such that ⇒| C A t | 1 1. gts→ 0 or 2. Ω C ≫Ω p where Ω P is the cut-off frequency of gts ps. Riemann – Lebesgue lemma

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ONE CAN HAVE DIFFERENT CONTROL FUNCTION OR PULSE SEQUENCES But effectiveness of control only depends on the integral h 0 s ∫ ′ s d ′ s Ωt− s e.g. Bang-Bang control Ω Φ 0 τ /Δ ht

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BANG-BANG PULSES AND NOISE PROTECTED MEMORY E1: ρ C A At At + ... C A 0 1 for example a time-indepedent state At i.e. ⎯ → ⎯ a 0 + b 1 using the even interval pulse control or whatever pulses even noise Or a QSD equation zψ C A At + ... where At i.e. ⎯ → ⎯ a 0 + b 1

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C A te −Lt Ω+ Φ 0 τ /Δ ≫ω d where Ω system frequency. ΔΦ 0 τ pulse avarage frequency ω d bath cutoff. Regular Fast Pulse Control

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CONTROL WITH RANDOMNESS AND EVEN BY NOISE

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Noise-Induced Adiabaticity SCHRÖDINGER EQUATION AND EIGEN-EQUATION H t E n t E n t E n t i∂ t ψ t H t ψ t where ψ t ψ n t n ∑ E n t Instantaneous Eigenstate Dynamical Phase ∂ t ψ m −iE m ψ m − E m E n ψ n n ∑

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STANDARD ADIABATIC CONDITION E m H E n ≪ E n − E m ψ m t 1 H m≠n −i E m E n THE HAMILTONIAN in ROTATING REPRESENTATION

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ONE-COMPONENT DYNAMICAL EQUATION. tracing footprint of target state At E 0 t C A p ψ 0 ∂ t ψ 0 t − dsg ts 0 t ∫ ψ 0 s General Adiabatic necessary Condition dsg ts ψ 0 s 0 t ∫ 0 TWO-LEVEL SYSTEM g ts − E 0 t E 1 t E 1 s E 0 s e iE− E 1 E 1 − E 0 E 0 s t ∫ d ′ s ht E E 0 − E 1

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Adiabatic and non-adiabatic passages H t J 0 t T σ x + 1− t T ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ σ z ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ g ts 4e i E ′ s d ′ s s t ∫ T 2 E 2 t E 2 s H 0 J 0 σ z ⇒ 0 H T J 0 σ x ⇒ 1 2 0 − 1 . T 5 / J 0 The fast-varying factor. It can be noise 0 0.2 0.4 0.6 0.8 1 0.75 0.8 0.85 0.9 0.95 1 t/T | ψ 0 | T1/J 0 T2/J 0 T3/J 0 T4/J 0 T5/J 0 T6/J 0 T7/J 0

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NOISE INDUCED ADIABATICITY J 0 ⇒ J 0 + c Jt c plays the same role as ht Continuous biased Poissonian white shot noise H ⇒ 1+ c Jt J 0 ⎡ ⎣ ⎤ ⎦ H

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EXAMPLE OF STRENGTH NOISE H J 0 σ z +cJt σ z In lab framework the model could be First: σ in rotating framework frequency ω Second: noise in rotating framework. H J 0 +cJtcosωtσ x +sinωtσ y σ z ⇒σ x

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NOISE-INDUCED ADIABATICITY 0 0.2 0.4 0.6 0.8 1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t/T | ψ 0 | J0 J0.1J 0 J0.5J 0 JJ 0 0 0.2 0.4 0.6 0.8 1 −1 0 1 t/T Slow varying function Fast varying Noise function g ts 4e i J 0 k ′ s ⎡ ⎣ ⎤ ⎦ d ′ s s t ∫ T 2 k 2 t k 2 s e i c J ′ s k ′ s ⎡ ⎣ ⎤ ⎦ d ′ s s t ∫

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APPLICATION: SPEED UP QUANTUM ADIABATIC ALGORITHM 3SAT example H J 0 1−t /T 0 H B + t /T 0 H P Eigenstates E H P ⇒solution We use J 0 ⇒ J 0 +c Jt T 0 160 in the abiabatic regime J big enough T 0 can be as short as we wish T 0 5

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ψ 0

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CONCLUSION  We first derive a one-component integro- differential equation by P-Q partitioning.  We introduce an LEO to separate P-space from Q-space and approximate LEO  All given quantum paths may be realized in terms of fast-varying signals even noise signals.  We find that the effectiveness of LEOs exclusively depends on the integral of the pulse sequence or control function in the time domain which has been missing for a long time.

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Thanks for attentions   Many thanks to organizers   One joint postdoc opening

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