Calculating transition amplitudes by variational quantum eigensolvers
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ü VQD can accurately simulate excited states on a quantum computer but cannot calculate transition amplitudes between states ü We proposed a method to calculate transition amplitudes between two orthogonal states obtained from VQD in a hardware-friendly manner using variational algorithms ü This allows calculation of important physical properties like oscillator strengths that require transition amplitudes, expanding the capabilities of VQD for excited state simulations on NISQ devices
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![𝑅 [Å] 𝑅 [Å]𝑅 [Å]
Calculating transition amplitudes by variational quantum eigensolvers
C71.00241 : Implementation of excited state energy and its analytical derivatives for photochemical reaction simulations on NISQ devices
Yohei Ibe1, Yuya O. Nakagawa1,Takahiro Yamamoto1, Kosuke Mitarai2, 1, Tennin Yan1, Qi Gao3, Takao Kobayashi3
1QunaSys Inc., 2Osaka University, 3Mitsubishi Chemical Corp.
1. Methods for excited states based on VQE
ü Simulating excited states of molecules is of great concern in photophysics & photochemistry,
and there are several methods based on a quantum algorithm, Variational Quantum Eigensolver (VQE)
ü Among them, VQD method is remarkably accurate; however, there are no methods to calculate
transition amplitudes (i.e., off-diagonal matrix elements) between eigenstates obtained by the VQD
ü We propose such a method feasible on NISQ devices and demonstrate on a sampling simulator
Summary
3. Calculating transition amplitudes
ü VQD is an accurate way to simulate excited states on a quantum
computer
ü We proposed a method to calculate transition amplitudes
between two orthogonal states in a hardware-friendly manner,
which is applicable (not only) for the VQD
ü This work enlarges the possibility of the VQD and advances the
field of excited states calculations on a quantum device
4. Sampling simulation with shot noise
5. Conclusion
Appendix: Comparison of ansatz
LiH
𝑅
Li H
STO-3G
(4e, 6o)
(12 qubits)
Method for
excited states
Accuracy
Easy to calculate
transition amplitudes?
SSVQE Fair Yes
MCVQE Intermediate Yes
VQD Excellent No Yes (This work)
Diazene (N2H2)
2. Comparison of methods by noiseless simulation
Potential energy curve Error from exact calc.
O. Higgott et al., Quantum 3, 159 (2019)R. Parrish et al.,
Phys. Rev. Lett. 122, 230401 (2019)
K. Nakanishi et al.,
Phys. Rev. Research 1, 033062 (2019)
ℒ(𝜽) = *
+,-
.
𝑤+⟨𝜑+ 𝑈3
𝜽 𝐻𝑈 𝜽 𝜑+⟩
(𝑤-> 𝑤7 > ⋯ > 0)
ℒ(𝜽) = *
+,-
.
⟨𝜑+ 𝑈3
𝜽 𝐻𝑈 𝜽 𝜑+⟩
1. Obtain ground state E; with VQE
2. Execute VQE routine to minimize
ℒ- 𝜃 = 𝜓 𝜃 𝐻 𝜓 𝜃 + 𝑤- ⟨E;|𝜓 𝜃 ⟩ 7
→ Obtain first excited state E- = 𝜓 𝜃-
∗
3. Repeat for higher excited states
SSVQE
Subspace-Search VQE
MCVQE
Multistate-Contracted VQE
VQD
Variational Quantum Deflation
: physical quantity (Hermitian op.), where .
Following equality holds for transition amplitude :
Each term can be measured on real devices
Calculation setups
• Molecular structures: several points along minimum energy
path between S2 Franck-Condon (cis/trans) & S2 minimum
• Calculation level: 6-31G*/CASCI(3o, 4e), 8 qubits
• Ansatz: RSP ansatz (D=20, see Appendix)
• High-speed simulator Qulacs [http://qulacs.org/] is used
Enables calculation of transition amplitudes with VQD
Unable to measure
as it is (A is not unitary)
Definition of oscillator strength 𝑓+B
: 𝛼-coordinate of the 𝑙-th electron
where
: electric dipole moment operator
in atomic units
LiH
𝑅
Li H
STO-3G
CASCI(2e, 2o)
(2 qubits)
Potential energy curve & error Oscillator strength
RY ansatz
ü VQD can generate excited states the most accurately
ü SSVQE and MCVQE can readily calculate transition amplitudes
ü However, VQD has no known methods to calculate transition
amplitudes on real NISQ devices
ü Transition amplitudes are required for calculating
various physical quantities (e.g., oscillator strengths)
(𝐴: Hermitian op.)
Calculation setups
• Calculation level: 6-31G*/CASCI(2o, 2e), 2 qubits (parity mapping is used to reduce the number of qubits)
• Ansatz: RY ansatz (D=2)
• Two options for the optimization routine for this experiment
Blue dots: use sampling simulator in the whole process (including optimization routine)
Orange dots: use sampling simulator only for calculating oscillator strengths (parameters are optimized
with the noiseless simulator), still using our proposed method
Using RSP ansatz (D=10, see Appendix)
J. T. Seeley et al., J. Chem. Phys. 137, 224109 (2012)
(Available in Qiskit Aqua v0.6.4)
Unitary operators
Assuming
Hardware-efficient ansatz
𝑈FGGHI 𝜽 = exp 𝑇HI(𝜽) − 𝑇HI
3
(𝜽)
𝑇HI 𝜽 = *
+:PQQRSTUV
W:XTYZR[
𝜃+
W
𝑐W
3
𝑐+
+ *
+B:PQQRSTUV
W^:XTYZR[
𝜃+B
W^
𝑐W
3
𝑐^
3
𝑐+ 𝑐B
A. Kandala et al., Nature 549, 242 (2017) A. Peruzzo et al., Nat. Comm., 5,4213 (2014)
×𝐷
UCC-SD ansatz RSP ansatz
(real-valued symmetry preserving ansatz)
𝑅 [Å] 𝑅 [Å]
𝑅 [Å]
𝑅 [Å]
Energy[Ha]
For original symmetry-preserving ansatz, see
P. Barkoutsos et al., Phys. Rev. A 98, 022322 (2018)
Energy[Ha]
Energy[Ha]
Molecule: LiH(4e, 6o)
Basis set: STO-3G
# of qubits: 12
depth D = 8
Molecule: LiH(4e, 6o)
Basis set: STO-3G
# of qubits: 12
Molecule: LiH(4e, 6o)
Basis set: STO-3G
# of qubits: 12
depth D = 20
For results on azobenzene, see our paper on arXiv!
arXiv:2002.11724
automatically generates multiple states finally, diagonalize on a classical computer](https://image.slidesharecdn.com/aps2020postergcp-200303053025/85/Calculating-transition-amplitudes-by-variational-quantum-eigensolvers-1-320.jpg)
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2) Rewrite the differential equation for a mechanical system where damping force is proportional to velocity squared and spring force is proportional to displacement squared.
3) Derive a state model and discuss if frequency tools can analyze this nonlinear system.
2. The document asks to:
1) Derive an expression for thermal resistance using the law of thermal conduction.
2) Identify the through and across variables for a chemical system based on concentration and flux, write the equation relating them, and the equation for dissipated power/stored energy.
PFN Summer Internship 2021 / Kohei Shinohara: Charge Transfer Modeling in Neu...
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Hh model(me)
The Hodgkin-Huxley model is a mathematical model that describes the generation and propagation of electrical signals in neurons. It consists of nonlinear differential equations that describe the voltage-dependent ionic currents that flow across the neuronal membrane. The model includes separate voltage-gated ion channels for sodium and potassium ions that allow the currents to change over time based on the membrane potential. The model can be used to simulate the membrane potential over time in response to an applied electric current stimulus.
Methods available in WIEN2k for the treatment of exchange and correlation ef...
This document summarizes methods available in the WIEN2k software for treating exchange and correlation effects beyond semilocal density functional theory. It discusses the semilocal generalized gradient approximation and meta-GGA functionals, the modified Becke-Johnson potential for improving band gaps, dispersion correction methods, and on-site corrections like DFT+U and hybrid functionals for strongly correlated materials. Input parameters and keywords for selecting these methods in the WIEN2k code are also outlined.
Vibration energy harvesting under uncertainty
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Lecture6
Lecture 6: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
Post_Number Systems_8.1.2
1. The document discusses number systems and their use in physical and computational sciences. It focuses on exploring natural events in Fibonacci number space and the medical sciences.
2. It establishes a mathematical definition for natural events that should correlate to the Bernoulli base of natural logarithms. A mathematical description of nature should only be accurate if the number system complies with the natural conditions of the number one.
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THE RESEARCH OF QUANTUM PHASE ESTIMATION ALGORITHM
This document discusses phase estimation in quantum computing. It begins by introducing quantum Fourier transforms and how they are important for algorithms like Shor's algorithm. It then describes the phase estimation algorithm in detail, including how it uses two registers to estimate the phase of a quantum state and how the inverse quantum Fourier transform improves this estimate. Simulation results are presented that show the probability distribution of the estimated phase converging to the true value and how the probability of success increases with more qubits while computational costs rise polynomially. The paper concludes that the optimal number of qubits balances high success probability and low costs for phase estimation.
1789 1800
This document describes the construction and selection of single sampling quick switching variables systems for given control limits that involve minimum sum of risks. It provides the procedure for finding the single sampling quick switching variables system that has the minimum sum of producer's and consumer's risk for a specified acceptable quality level and limiting quality level. A table is constructed that can be used to select a quick switching variables sampling system for given values of AQL and LQL that has the minimum sum of risks. The document also discusses how to design a quick switching variables sampling system with an unknown standard deviation that involves minimum sum of risks.
1789 1800
This document describes the construction and selection of single sampling quick switching variables systems for given control limits that involve minimum sum of risks. It provides the procedure for finding the single sampling quick switching variables system that has the minimum sum of producer's and consumer's risk for a specified acceptable quality level and limiting quality level. A table is constructed that can be used to select a quick switching variables sampling system for given values of AQL and LQL that has the minimum sum of risks. The document also discusses how to design a quick switching variables sampling system with an unknown standard deviation that involves minimum sum of risks.
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Science Journals,
Extraction of photovoltaic generator parameters through combination of an an...
In the present work, we propose an improved method based on a combination of an analytical and iterative approach to extract the photovoltaic (PV) module parameters using the measured current-voltage characteristics and the simple diode model. First, we calculate the series resistance using a set of analytical formulas for the base values of the three current-voltage curves. Then, the three other parameters are analytically expressed as functions of serial resistance and ideality factor based on the linear least-squares method. Finally, the ideality factor is calculated applying an iterative algorithm to minimize the normalized root mean square error (NRMSE) value. The proposed method was validated with a real experimental set of two PV generators, which showed the best fit to the I-V curve. Moreover, the proposed method needs only the initial value of the ideality factor.
591 adamidis
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Tuning of PID, SVFB and LQ Controllers Using Genetic Algorithms
Tuning of PID, SVFB and LQ Controllers Using Genetic AlgorithmsInternational Journal of Engineering Inventions www.ijeijournal.com
1) The document describes using genetic algorithms to tune PID, state variable feedback, and LQR controllers for balancing an inverted pendulum on a cart.
2) It presents the mathematical model of the inverted pendulum system and linearizes the model.
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A quantum computation problem is discussed in this paper. Many new features that make
quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform
algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is
presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is
analysed. The probability distribution of the measuring result of phase value is presented and
the computational efficiency is discussed.
COMPUTATIONAL PERFORMANCE OF QUANTUM PHASE ESTIMATION ALGORITHM
A quantum computation problem is discussed in this paper. Many new features that make quantum computation superior to classical computation can be attributed to quantum coherence
effect, which depends on the phase of quantum coherent state. Quantum Fourier transform algorithm, the most commonly used algorithm, is introduced. And one of its most important
applications, phase estimation of quantum state based on quantum Fourier transform, is presented in details. The flow of phase estimation algorithm and the quantum circuit model are
shown. And the error of the output phase value, as well as the probability of measurement, is analysed. The probability distribution of the measuring result of phase value is presented and the computational efficiency is discussed.
Indoor Heading Estimation
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This document describes a new rotor balancing method that uses an optimization technique called genetic algorithm to calculate correction masses without requiring test runs. It introduces the new method and compares it to the traditional influence coefficient method. The new method calculates theoretical unbalance responses and measures original unbalance vibrations to optimize the correction masses using genetic algorithm in order to minimize residual vibrations at selected speeds and locations. Both simulation and experimental results show the new method can effectively reduce residual vibrations.
Optical sensing techniques and signal processing 5
This document discusses frequency analysis of optical imaging systems. It begins with a generalized treatment of imaging systems and defines the amplitude transfer function and impulse response. For coherent systems, the image is a convolution of the ideal image and amplitude impulse response. For incoherent systems, the image intensity is a convolution with the intensity impulse response, which is the squared amplitude impulse response. It then examines the frequency responses of coherent and incoherent systems, defining the amplitude transfer function and optical transfer function respectively.
Electron impact TCS of HCl and HBr .ppt
This document summarizes the calculation of total electron impact cross sections for HCl and HBr over a wide energy range from 0.1 eV to 2 keV. Two formalisms were required to cover this range - the R-matrix method was used from 0.01 eV to the ionization threshold, and the SCOP (Spherical Complex Optical Potential) method was used from the ionization threshold to 2 keV. The results from the two methods were consistent at the transition energy and showed good agreement with available experimental data. Total cross sections contain information on various scattering processes and this methodology could be applied to calculate cross sections for other molecular systems where experiments are difficult.
Proceeding - PANIC
The document summarizes a proposed experiment called TREK at the J-PARC facility in Japan. The experiment aims to search for violations of time reversal symmetry by measuring the transverse muon polarization (PT) in stopped kaon (K+) decays with high precision (10-4). An active polarimeter will be used to stop muons from K+ decays and detect the asymmetry in positron emissions to extract the T-violating parameter PT, which is predicted to be very small in the Standard Model but could indicate new physics if measured to be non-zero. The upgraded TREK detector design, including an active scintillating fiber target, electromagnetic calorimeter, tracking chambers, and active muon polarimeter, is described
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1) The document discusses curvelet transformation and its application to object tracking. Curvelet transformation is a multiscale directional transform that can efficiently represent objects with curved edges using only a small number of coefficients.
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wep153
This summary provides the key details about simulation results of a feedback control system to damp electron cloud instabilities in the CERN SPS:
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- With a gain of 0.2, the feedback was able to reduce vertical instabilities and prevent emittance growth for an electron density of 1×1012 m-3.
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Destructive Fermi Resonance in 2D Harmoinc Oscillator
1) The document describes a computer simulation of a 2D harmonic oscillator model with and without coupling between the modes.
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Calculating transition amplitudes by variational quantum eigensolvers
- 1. 𝑅 [Å] 𝑅 [Å]𝑅 [Å] Calculating transition amplitudes by variational quantum eigensolvers C71.00241 : Implementation of excited state energy and its analytical derivatives for photochemical reaction simulations on NISQ devices Yohei Ibe1, Yuya O. Nakagawa1,Takahiro Yamamoto1, Kosuke Mitarai2, 1, Tennin Yan1, Qi Gao3, Takao Kobayashi3 1QunaSys Inc., 2Osaka University, 3Mitsubishi Chemical Corp. 1. Methods for excited states based on VQE ü Simulating excited states of molecules is of great concern in photophysics & photochemistry, and there are several methods based on a quantum algorithm, Variational Quantum Eigensolver (VQE) ü Among them, VQD method is remarkably accurate; however, there are no methods to calculate transition amplitudes (i.e., off-diagonal matrix elements) between eigenstates obtained by the VQD ü We propose such a method feasible on NISQ devices and demonstrate on a sampling simulator Summary 3. Calculating transition amplitudes ü VQD is an accurate way to simulate excited states on a quantum computer ü We proposed a method to calculate transition amplitudes between two orthogonal states in a hardware-friendly manner, which is applicable (not only) for the VQD ü This work enlarges the possibility of the VQD and advances the field of excited states calculations on a quantum device 4. Sampling simulation with shot noise 5. Conclusion Appendix: Comparison of ansatz LiH 𝑅 Li H STO-3G (4e, 6o) (12 qubits) Method for excited states Accuracy Easy to calculate transition amplitudes? SSVQE Fair Yes MCVQE Intermediate Yes VQD Excellent No Yes (This work) Diazene (N2H2) 2. Comparison of methods by noiseless simulation Potential energy curve Error from exact calc. O. Higgott et al., Quantum 3, 159 (2019)R. Parrish et al., Phys. Rev. Lett. 122, 230401 (2019) K. Nakanishi et al., Phys. Rev. Research 1, 033062 (2019) ℒ(𝜽) = * +,- . 𝑤+⟨𝜑+ 𝑈3 𝜽 𝐻𝑈 𝜽 𝜑+⟩ (𝑤-> 𝑤7 > ⋯ > 0) ℒ(𝜽) = * +,- . ⟨𝜑+ 𝑈3 𝜽 𝐻𝑈 𝜽 𝜑+⟩ 1. Obtain ground state E; with VQE 2. Execute VQE routine to minimize ℒ- 𝜃 = 𝜓 𝜃 𝐻 𝜓 𝜃 + 𝑤- ⟨E;|𝜓 𝜃 ⟩ 7 → Obtain first excited state E- = 𝜓 𝜃- ∗ 3. Repeat for higher excited states SSVQE Subspace-Search VQE MCVQE Multistate-Contracted VQE VQD Variational Quantum Deflation : physical quantity (Hermitian op.), where . Following equality holds for transition amplitude : Each term can be measured on real devices Calculation setups • Molecular structures: several points along minimum energy path between S2 Franck-Condon (cis/trans) & S2 minimum • Calculation level: 6-31G*/CASCI(3o, 4e), 8 qubits • Ansatz: RSP ansatz (D=20, see Appendix) • High-speed simulator Qulacs [http://qulacs.org/] is used Enables calculation of transition amplitudes with VQD Unable to measure as it is (A is not unitary) Definition of oscillator strength 𝑓+B : 𝛼-coordinate of the 𝑙-th electron where : electric dipole moment operator in atomic units LiH 𝑅 Li H STO-3G CASCI(2e, 2o) (2 qubits) Potential energy curve & error Oscillator strength RY ansatz ü VQD can generate excited states the most accurately ü SSVQE and MCVQE can readily calculate transition amplitudes ü However, VQD has no known methods to calculate transition amplitudes on real NISQ devices ü Transition amplitudes are required for calculating various physical quantities (e.g., oscillator strengths) (𝐴: Hermitian op.) Calculation setups • Calculation level: 6-31G*/CASCI(2o, 2e), 2 qubits (parity mapping is used to reduce the number of qubits) • Ansatz: RY ansatz (D=2) • Two options for the optimization routine for this experiment Blue dots: use sampling simulator in the whole process (including optimization routine) Orange dots: use sampling simulator only for calculating oscillator strengths (parameters are optimized with the noiseless simulator), still using our proposed method Using RSP ansatz (D=10, see Appendix) J. T. Seeley et al., J. Chem. Phys. 137, 224109 (2012) (Available in Qiskit Aqua v0.6.4) Unitary operators Assuming Hardware-efficient ansatz 𝑈FGGHI 𝜽 = exp 𝑇HI(𝜽) − 𝑇HI 3 (𝜽) 𝑇HI 𝜽 = * +:PQQRSTUV W:XTYZR[ 𝜃+ W 𝑐W 3 𝑐+ + * +B:PQQRSTUV W^:XTYZR[ 𝜃+B W^ 𝑐W 3 𝑐^ 3 𝑐+ 𝑐B A. Kandala et al., Nature 549, 242 (2017) A. Peruzzo et al., Nat. Comm., 5,4213 (2014) ×𝐷 UCC-SD ansatz RSP ansatz (real-valued symmetry preserving ansatz) 𝑅 [Å] 𝑅 [Å] 𝑅 [Å] 𝑅 [Å] Energy[Ha] For original symmetry-preserving ansatz, see P. Barkoutsos et al., Phys. Rev. A 98, 022322 (2018) Energy[Ha] Energy[Ha] Molecule: LiH(4e, 6o) Basis set: STO-3G # of qubits: 12 depth D = 8 Molecule: LiH(4e, 6o) Basis set: STO-3G # of qubits: 12 Molecule: LiH(4e, 6o) Basis set: STO-3G # of qubits: 12 depth D = 20 For results on azobenzene, see our paper on arXiv! arXiv:2002.11724 automatically generates multiple states finally, diagonalize on a classical computer