Variational Quantum Algorithm for Markovian Open Quantum Systems

VARIATIONAL QUANTUM ALGORITHM FOR
MARKOVIAN OPEN QUANTUM SYSTEMS
N. Yoshioka (U. Tokyo), Y. O. Nakagawa (QunaSys)
K. Mitarai, and K, Fujii (Osaka U.)
arXiv: 1908.09836
APS March meeting 2020
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eD V
V ⇤
|ρ⟩
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2
Introduction: Open quantum system
Taken from Garcia-Ripoll et al., J. Phys B (’05)
Environment
- Usually macroscopic
- Usually stable, act as heat bath etc.
e.g.) Room temperature obj.
e.g.) Measurement backaction
interact
ion qubits
System
- Microscopic
e.g.) Superconducting qubits
e.g.) Cold atom, Trapped ions
- Coherent/entangled etc.
CCD
camera

3
Introduction: Open quantum system
Taken from Garcia-Ripoll et al., J. Phys B (’05)
Environment
- Usually macroscopic
- Usually stable, act as heat bath etc.
e.g.) Room temperature obj.
e.g.) Measurement backaction
interact
ion qubits
System
- Microscopic
e.g.) Superconducting qubits
e.g.) Cold atom, Trapped ions
- Coherent/entangled etc. When environment is traced out,
system with dissipation is left.
CCD
camera

4
Time evolution and eigenstates in isolated/open systems
Quantum master equation assumes
1. time-homogeneous, Markovian process
2. CPTP (completely-Positive and Trace-Preserving)
System
Time evolution
equation
Eigenstates Remarks
Isolated
Open
i~
d| i
dt
= ˆH| i
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Hamiltonian
Pure
dˆ⇢(t)
dt
= Lˆ⇢(t)
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Liouvillian
Mixed
Ground/excited states
Steady state
Decaying modes
Lˆ⇢ = 0
Lˆ⇢ = ˆ⇢
ˆH| i = E| i
- Application of NISQ to
- Numerous investigations done
- Quantum chemistry
- Finance
- Important to understand
- transports in quantum systems
- non-equilibrium topo. phase
- Few algorithms,
No demonstration in NISQ
Target today

5
Our proposal: dissipative-system VQE (dVQE) algorithm
2. Define cost function
arg min
⇢
h⇢| ˆL† ˆL|⇢i
h⇢|⇢i<latexit sha1_base64="TGQB/+cZOWhBLC8vU1L1/GFbI8M=">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</latexit>
Execute optimization
|⇢✓i = [V (✓v) ⌦ V ⇤
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1. Encode mixed state in quantum circuit
⇥
NY
n=1
CNOTn,n+N
!
eD(✓d)|0i
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3. Measure observables
V
|q1i
|q2i
|q3i
|q0
1i
|q0
2i
|q0
3i|q00
3 i
|q00
2 i
|q00
1 i
hAi = Tr[ˆ⇢ Â] =
X
q
qh q| Â| qi
=
X
q
qhq| ˆV † Â ˆV |qihAi = Tr[ˆ⇢ Â] =
X
q
qh q| Â| qi
=
X
qhq| ˆV † Â ˆV |qi

1. Encoding mixed state into variational quantum circuit
Assured in actual measurement
(Step 3)
Physical requirements for density matrices (general mixed state)
(I) ⇢†
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(II) h |⇢| i 0,8
| i 2 H<latexit sha1_base64="I1IwiNBK0Z3V/tgyBSE4/7yZ3I4=">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</latexit>
(III) Tr[⇢] = 1<latexit sha1_base64="fRkFC7x1HJDVxZUO8ykWK3e0QrU=">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</latexit>
Encoded in ansatz
}<latexit sha1_base64="WlNroWubmpIAlBmPf6JE32RMyUo=">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</latexit>
Requirements (I),(II) in matrix/vector representation
(Hermiticity)
(Positivity)
(Unit trace)
Matrix rep. Vector rep.
6
⇢ =
X
ij
⇢ij|iihj| 7! |⇢i =
X
ij
⇢ij
C
|iiP ⌦ |jiA
<latexit sha1_base64="H0SrQ2QD5F42uQeERADoKsrdSw8=">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</latexit>
⇢ = V DV †
D = diag ({ q})<latexit sha1_base64="Agw+WrqwuZQWyqRN3JCBFvP0NDk=">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</latexit>
(I), (II) is equivalent to rewriting
where ∀
λ ≥ 0
⇢ =
X
ij
⇢ij|iihj| 7! |⇢i =
X
ij
⇢ij
C
|iiP ⌦ |jiA
|⇢i = [V ⌦ V ⇤
] |Di<latexit sha1_base64="v23kTsKwumtZuGK9PT8OGjKoHRo=">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</latexit>
States are mapped as
|Di = eD|0i =
X
q
q
C
|qiP ⌦ |qiA
7!
7!
Choi
isomorphism
C: normalization
const.

Physical requirements for density matrices (general mixed state)
(I) ⇢†
= ⇢<latexit sha1_base64="7TfBV7/uCDpc3+k1tdGywq2IlQc=">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</latexit>
(II) h |⇢| i 0,8
| i 2 H<latexit sha1_base64="I1IwiNBK0Z3V/tgyBSE4/7yZ3I4=">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</latexit>
(III) Tr[⇢] = 1<latexit sha1_base64="fRkFC7x1HJDVxZUO8ykWK3e0QrU=">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</latexit>
Encoded in ansatz
}<latexit sha1_base64="WlNroWubmpIAlBmPf6JE32RMyUo=">AAACZXichVHLSgMxFD0dX7W+6gMRXCgWxVXJVEFxJbpx2VargorMjKmGzouZtFCLPyBu1YUrBRHxM9z4Ay76BSIuK7hx4Z3pgKioNyQ5Obnn5iTRXVP4krF6TGlpbWvviHcmurp7evuS/QPrvlP2DF4wHNPxNnXN56aweUEKafJN1+OapZt8Qy8tB/sbFe75wrHXZNXlO5a2b4uiMDRJVH77aDeZYmkWxvhPoEYghSiyTvIG29iDAwNlWOCwIQmb0OBT24IKBpe4HdSI8wiJcJ/jCAnSlimLU4ZGbInGfVptRaxN66CmH6oNOsWk7pFyHJPskd2yBntgd+yZvf9aqxbWCLxUadabWu7u9h2PrL79q7Joljj4VP3pWaKI+dCrIO9uyAS3MJr6yuF5Y3UhP1mbYlfshfxfsjq7pxvYlVfjOsfzF0jQB6jfn/snWM+k1Zl0JjebWlyKviKOUUxgmt57DotYQRYFOreIE5ziLPak9ChDynAzVYlFmkF8CWXsA99limc=</latexit>
Requirements (I),(II) in matrix/vector representation
(Hermiticity)
(Positivity)
(Unit trace)
7
Vector rep.
⇢ij|iihj| 7! |⇢i =
X
ij
⇢ij
C
|iiP ⌦ |jiA
|⇢i = [V ⌦ V ⇤
Choi isomorphism maps states to
|Di = eD|0i =
X
q
q
C
|qiP ⌦ |qiA
Generate
{λq}
Entangle as
|q⟩ ⊗ |q⟩
Basis rot.
eD V
V ⇤
Our ansatz
|ρ⟩
1. Encoding mixed state into variational quantum circuit
is chosen to produce positive wavefunciton (see next page)˜D
Assured in actual measurement
(Step 3)

Examples of variational quantum circuits
Circuit for eigenvalues :{λq}
eD<latexit sha1_base64="QpY29Ife68aPsKK1Jsj5c+HPCo8=">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</latexit>
V<latexit sha1_base64="Nm5exm1apqA9q8w9hBxWdE4GLH4=">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</latexit>
V ⇤
<latexit sha1_base64="DZjP0U4CSK4n1SYC7A1f2KAjT4Q=">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</latexit>
|0iP<latexit sha1_base64="qiyqzRCvIXZuuyb3r5ZboH8/mUs=">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</latexit>
|Di<latexit sha1_base64="JB8kojTJdAWcgmAVMJAc/uWHV88=">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</latexit>
Ry<latexit sha1_base64="v2NhGacKRxSJu6JDCr5kq8J39po=">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</latexit>
{<latexitsha1_base64="ARaIdGOyud1ubRGqcGdCsuAk3Lw=">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</latexit>
(a) (b)⇥D1<latexit sha1_base64="NSL/CI71YIlWv+LgPeZFdeeH0+s=">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</latexit>
eD<latexit sha1_base64="QpY29Ife68aPsKK1Jsj5c+HPCo8=">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</latexit>
Circuit for basis transformation: V<latexit sha1_base64="Nm5exm1apqA9q8w9hBxWdE4GLH4=">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</latexit>
Rx<latexit sha1_base64="LiKptAD3Y7foGtV9xCI+6WzR6no=">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</latexit>
{<latexitsha1_base64="ARaIdGOyud1ubRGqcGdCsuAk3Lw=">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</latexit>
⇥D2<latexit sha1_base64="NZzT4Mvk10j9UgR+cS43CWp1LIA=">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</latexit>
e.g. hardware-efﬁcient ansatz
8
|⇢i = [V ⌦ V ⇤
|Di = eD|0i =
X
q
q
C
|qiP ⌦ |qiA
Our ansatz
λq ≥ 0
where angle θ of each Ry=exp(-iθY/2)) satisﬁes
so as to
0 ≤ θ ≤ π
λq ≥ 0

2. Define cost function — variational search of steady states
Im !
Re !
Decaying
modes
“Spiral”
Spectrum of ℒ#
Re(Λ)
Spectrum of ℒ#'ℒ#
Stationary
state
Real, non-negative
spectrum
Stationary state as kernel
This allows us to obtain steady state via
Cui et al. PRL (’15)
ˆL|⇢i = 0<latexit sha1_base64="QdqtvcXNKM5HnayycoOu8lK8S08=">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</latexit>
ˆL† ˆL|⇢i = 0<latexit sha1_base64="5GGYmsM86HPrc/9EXF+DZ/B7Mmo=">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</latexit>
9
Unitary evol. Non-unitary evol.
(e.g. for spontaneous emission, T1 effect)î = i<latexit sha1_base64="Mr9O8p+hYUWwzg4h5kHoHulBG70=">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</latexit>
Markovian master equation in Lindblad form
ˆD[î] = î ⌦ ˆ⇤
i
1
2
ˆ†
i
î ⌦ ˆ1 ˆ1 ⌦
1
2
ˆT
i
ˆ⇤
i
<latexit sha1_base64="YOaPKL04VqLlwQAIxyXdafMJNVA=">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</latexit>
ˆL|⇢(t)i = i
⇣
ˆH ⌦ ˆ1 ˆ1 ⌦ ˆHT
⌘
+
X
i
i
ˆD[î]
!
|⇢(t)i
<latexit sha1_base64="J5lqtFqpHedrFQg+24GzzB/1qSk=">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</latexit>
where
Steady
state
arg min
⇢
h⇢| ˆL† ˆL|⇢i
h⇢|⇢i<latexit sha1_base64="TGQB/+cZOWhBLC8vU1L1/GFbI8M=">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</latexit>

3. Measure Observables — efficient way
Naive way:
Vector Representation using 2N qubit
- Post-select “diagonal terms”  
- Exponentially inefficient
Our proposal:
Matrix Representation using N qubit
- Determine {λq} from sampling 
- Measure with initial bit q with weight λq
hAi = Tr[ˆ⇢ Â] =
X
q
qh q| Â| qi
=
X
q
qhq| ˆV † Â ˆV |qi
eD V
V ⇤
V
|q1i
|q2i
|q3i
⇥ qhq| ˆV † Â ˆV |qi
10
Sampling

Demonstration of our proposal in quantum Ising model
H =
1
2
X
i
z
i
z
i+1 + g
X
i
x
i
<latexit sha1_base64="lNAX3ml++ID3x1m/BGSLrvuJ7d0=">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</latexit>
c
(1)
i = i<latexit sha1_base64="Xi2HIpaRIdUGLxErU4eX70aF2VY=">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</latexit>
c
(2)
i = z
i<latexit sha1_base64="/XrcWjKP75O37Q9VAf44ZS1s9Yc=">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</latexit>
Hamiltonian: ZZ coupling and traverse-field
Dissipation: Local damping & dephasing
Infidelity
hXi,hYi,hZi<latexitsha1_base64="5xmvkpepEZyrf1DW4o8nac0f30Y=">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</latexit>
hXi<latexit sha1_base64="oC/y3gHzEfsy25zkjQcGk8/YVO4=">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</latexit>
hY i<latexit sha1_base64="hJWrEK4v8IJf6dDVlHIPrK7vs/E=">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</latexit>
hZi<latexit sha1_base64="S4WsArTCUG07OMGMKv19OOgak1c=">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</latexit>
Quantum simulation on real NISQ device
(Aspen-4, Rigetti, 2N=2 qubits)
with amplitudes γ1 =1, γ2 = 0.5
Model
Calculating magnetization curve and density matrix fidelity by dVQE
Infidelity
Numerical simulation, 2N=4 qubits
11
ˆD[î] = î ⌦ ˆ⇤
i
1
2
ˆ†
i
î ⌦ ˆ1 ˆ1 ⌦
1
2
ˆT
i
ˆ⇤
i
<latexit sha1_base64="YOaPKL04VqLlwQAIxyXdafMJNVA=">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</latexit>
circles: data of mag. 
dashed lines: exact mag.
triangles: 
infidelity btw. exact state

Summary
- VQE-based algorithm for steady states in Markovian open quantum system (dVQE)
Inﬁdelity
12
eD V
V ⇤
- Numerical/Experimental demonstration in dissipative quantum Ising model
- Applications to
dissipative phase exploration in condensed matter systems
are expected
transport properties in nano devices/molecular systems
Yoshioka, Nakagawa, Mitarai, and Fujii, arXiv: 1908.09836

14
Encoding mixed states into pure states
Choi isomorphism
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ij
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ij
⇢ij
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|iiP ⌦ |jiA
(<latexit sha1_base64="PAteIQ+iIH66HU6YuElYCXKbCX8=">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</latexit>
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7!<latexit sha1_base64="leyKFJh8UEpVDpFmqDf1Xd/Z2pY=">AAACanichVG7SgNBFD1Z3/GRqI2SJhgVq3BXBcVKtLH0lQeoyO46iYv7YncS0OAP2FkJWimIiJ9h4w9Y5BNEuwg2Ft5sFkRFvcPMnDlzz50zM7pnmYEkqseUtvaOzq7unnhvX/9AIjk4lA/cim+InOFarl/UtUBYpiNy0pSWKHq+0GzdEgX9YLm5X6gKPzBdZ1MeemLH1sqOWTINTTJV2LY1L5DubjJDWQoj/ROoEcggilU3eYNt7MGFgQpsCDiQjC1oCLhtQQXBY24HNeZ8Rma4L3CMOGsrnCU4Q2P2gMcyr7Yi1uF1s2YQqg0+xeLuszKNCXqkW2rQA93RE73/WqsW1mh6OeRZb2mFt5s4Gdl4+1dl8yyx/6n607NECfOhV5O9eyHTvIXR0lePzhobC+sTtUm6omf2f0l1uucbONVX43pNrF8gzh+gfn/unyA/nVVnsurabGZxKfqKbqQwhil+7zksYgWryIXuTnGOi9iLMqSMKqlWqhKLNMP4Esr4B9B3jK8=</latexit>
(<latexit sha1_base64="PAteIQ+iIH66HU6YuElYCXKbCX8=">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</latexit>
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ˆ⇢ =<latexit sha1_base64="g5kzesKzhZhFLd/Fk9wFvXWVi00=">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</latexit>
|⇢ii<latexit sha1_base64="MYnfLBkC0klSOJRY2D2p9cYAEVI=">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</latexit>
T<latexit sha1_base64="nyPloKqtmtOdehDkHlVs+eISDrg=">AAACZHichVG7SgNBFD1Z3/GRaBAEQcSgWIW7KihWQRvLRBMVVGR3HXXJvtidBDT4A9oqFlYKIuJn2PgDFvkBQSwj2Fh4s1kQFfUOM3PmzD13zszonmUGkqgWU1pa29o7Orvi3T29fYlk/8Bq4JZ9QxQN13L9dV0LhGU6oihNaYl1zxearVtiTS8tNvbXKsIPTNcpyANPbNnanmPumoYmmcoXtpNpylAYoz+BGoE0osi5yRtsYgcuDJRhQ8CBZGxBQ8BtAyoIHnNbqDLnMzLDfYEjxFlb5izBGRqzJR73eLURsQ6vGzWDUG3wKRZ3n5WjGKdHuqU6PdAdPdP7r7WqYY2GlwOe9aZWeNuJ46GVt39VNs8S+5+qPz1L7GIu9Gqydy9kGrcwmvrK4Xl9ZX55vDpBV/TC/i+pRvd8A6fyalznxfIF4vwB6vfn/glWpzLqdEbNz6SzC9FXdGIYY5jk955FFkvIocjnCpzgFGexJ6VHSSmDzVQlFmlS+BLKyAevSInX</latexit>
Map N qubit mixed state → 2N qubit pure state

Details of optimization — sequential minimal optimization
Nakanishi-Todo-Fujii method
- Use periodicity of cost function w.r.t. angles θ
- Sine-curve-like structure with periods:
Cost function landscape in 2N=4 qubits
Nakanishi, Fuji,&Todo, arXiv:1903.12166
Costfunction
(a) (b) (c)
Angle Angle Angle
R⇤
x<latexit sha1_base64="ItkksZTq/GyMe22G2u2ABh3Ps+M=">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</latexit>
R⇤
y<latexit sha1_base64="L2ISrqNjUiBy/rCpSX4+W0cXEqg=">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</latexit>
R⇤
R⇤
R⇤
R⇤
R⇤
R⇤
R⇤
R⇤
R⇤
R⇤
period, Ry2π(a)
15
R⇤
R⇤
R⇤
R⇤
- for ordinary rotation gates2π
- for C(M)-Rotation2M+1
π
- if appears M times2π/M θ
eD V
period, CRy4π(b)
Costfunction
(a) (b) (c)
Angle Angle Angle
R⇤
R⇤
R⇤
R⇤
R⇤
R⇤
R⇤
R⇤
y<latexit sha1_base64="L2ISrqNjUiBy/rCpSX4+W0cXEqg=">AAACaHichVFNLwNBGH66vutrcUBcGg0Rh2aKhDgJF0fKtk18NLtrWhPb3
|0i
|0i
|0i
|0i
Ry
Ry Ry
RxRy
R⇤
x
RxRy
R⇤
y
R⇤
xR⇤
y
|0i
|0i
|0i
|0i
V ⇤
period, Ry x 2π & 2π(c)
Costfunction
(a) (b) (c)
Angle Angle Angle
Ry
Ry Ry
RxRy
R⇤
x
RxRy
R⇤
y
R⇤
xR⇤
y
|0i
|0i
|0i
|0i
Ry
Ry Ry
RxRy
R⇤
x
RxRy
R⇤
y
R⇤
xR⇤
y
|0i
|0i
|0i
|0i
Ry
Ry Ry
RxRy
R⇤
x
RxRy
R⇤
y
R⇤
xR⇤
y
|0i
|0i
|0i
|0i

Gate error mitigation
Quantum simulation with redundancy
Rx(±⇡/2) ⌘ Rz(⇡) (Rx(±⇡/2))
E
Rz(⇡)<latexit sha1_base64="qctadoWto16wZmr6MF95+YiMmnA=">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</latexit>
CZ ⌘ CZE
<latexit sha1_base64="8TF5MWEXpM+HEuv47KrMvA5Wwuk=">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</latexit>
Heya et al. , arXiv:1904.08566
E 2 {1, 3, 5, ...}<latexit sha1_base64="bHZGDF1iluyZfqCcf4lsBtajniw=">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</latexit>
for odd
2. Sample cost function
Costfunction
Angle
E = 1<latexit sha1_base64="+H9ckQTcPlj6lz+266RNnOqRePw=">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</latexit>
E = 5<latexit sha1_base64="aiMf3jyHOz3qO77U6uP63xAj3ic=">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</latexit>
E = 3<latexit sha1_base64="ew58bEBQdgB7jHYmKSXf9P1bg4Y=">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</latexit>
Cost function (2N=2, real NISQ device)
Numerical simulation: depolarizing
1. Applying depolarizing channel for each k-qubit gates as
⇢ 7! ⇢0
= (1 pk) ⇢ +
4k
1
4k
pkI
<latexit sha1_base64="0juK13j8vJHJy4Efr2zifjF/8dg=">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</latexit>
rate
2. Sample cost function
3. Mitigate by linear extrapolation w.r.t. pk
3. Mitigate by linear extrapolation w.r.t ε
16
1. Encode redundancy in the circuit using equation
(a circuit parameter)

Variational Quantum Algorithm for Markovian Open Quantum Systems

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