linear response on a quantum computer · roggero & carlson linear response on a qc int - 15...
Post on 12-Mar-2020
3 Views
Preview:
TRANSCRIPT
Linear response on a Quantum Computer
Alessandro Roggero & Joseph Carlson (LANL)
figure from JLAB collab.
NPQI - ANL - 28 Mar, 2018
Inclusive cross section and the response function
xsection completely determined by response function
R(q, ω) =∑f
∣∣∣〈f |O(q)|0〉∣∣∣2 δ (ω − Ef + E0)
excitation operator O(q) specifies the vertex
Same structure not only in NP but also condensed matter, cold atoms,. . .
Extremely challenging classically for strongly correlated quantum systems
limited to small systemsreliant on approximations that are difficult to control (efficiently)
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 1 / 3
Inclusive cross section and the response function
xsection completely determined by response function
R(q, ω) =∑f
∣∣∣〈f |O(q)|0〉∣∣∣2 δ (ω − Ef + E0)
excitation operator O(q) specifies the vertex
Same structure not only in NP but also condensed matter, cold atoms,. . .
Extremely challenging classically for strongly correlated quantum systems
limited to small systemsreliant on approximations that are difficult to control (efficiently)
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 1 / 3
Response functions on a Quantum Computeruse time correlation functions (Terhal&DiVincenzo(2000), Ortiz et al. (2001))
Ingredients for response calculation in frequency spacean oracle that prepares the ground state (QAA, VQE, Spectral Combing, . . . )an oracle for time evolution (Berry et al. (2015),Hao Low et al. (2016))
an oracle that prepares |E〉 = O(q)|0〉 (Roggero & Carlson (in prep.))
By performing quantum phase estimation (Kitaev(1996), Abrams&Lloyd(1999))with M ancilla qubits we will measure frequency ν with probability:
P (ν) =∑f
|〈f |E〉|2 δM (ν − Ef + E0)
finite width approximation of R(q, ω)
need only M ∼ log2 (1/∆ω) ancillaeevolution time t ∼ Poly(sys.size)/∆ω
Roggero & Carlson (in prep.)
101
102
103
104
105
106
107
108
Number of iterations
10-5
10-4
10-3
10-2
10-1
100
max
imu
m e
rro
r
M = 8
M = 12
~N-0.5
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 2 / 3
Response functions on a Quantum Computeruse time correlation functions (Terhal&DiVincenzo(2000), Ortiz et al. (2001))
Ingredients for response calculation in frequency spacean oracle that prepares the ground state (QAA, VQE, Spectral Combing, . . . )an oracle for time evolution (Berry et al. (2015),Hao Low et al. (2016))an oracle that prepares |E〉 = O(q)|0〉 (Roggero & Carlson (in prep.))
By performing quantum phase estimation (Kitaev(1996), Abrams&Lloyd(1999))with M ancilla qubits we will measure frequency ν with probability:
P (ν) =∑f
|〈f |E〉|2 δM (ν − Ef + E0)
finite width approximation of R(q, ω)
need only M ∼ log2 (1/∆ω) ancillaeevolution time t ∼ Poly(sys.size)/∆ω
Roggero & Carlson (in prep.)
101
102
103
104
105
106
107
108
Number of iterations
10-5
10-4
10-3
10-2
10-1
100
max
imu
m e
rro
r
M = 8
M = 12
~N-0.5
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 2 / 3
Response functions on a Quantum Computeruse time correlation functions (Terhal&DiVincenzo(2000), Ortiz et al. (2001))
Ingredients for response calculation in frequency spacean oracle that prepares the ground state (QAA, VQE, Spectral Combing, . . . )an oracle for time evolution (Berry et al. (2015),Hao Low et al. (2016))an oracle that prepares |E〉 = O(q)|0〉 (Roggero & Carlson (in prep.))
By performing quantum phase estimation (Kitaev(1996), Abrams&Lloyd(1999))with M ancilla qubits we will measure frequency ν with probability:
P (ν) =∑f
|〈f |E〉|2 δM (ν − Ef + E0)
finite width approximation of R(q, ω)
need only M ∼ log2 (1/∆ω) ancillaeevolution time t ∼ Poly(sys.size)/∆ω
Roggero & Carlson (in prep.)
101
102
103
104
105
106
107
108
Number of iterations
10-5
10-4
10-3
10-2
10-1
100
max
imu
m e
rro
r
M = 8
M = 12
~N-0.5
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 2 / 3
Response functions on a Quantum Computeruse time correlation functions (Terhal&DiVincenzo(2000), Ortiz et al. (2001))
Ingredients for response calculation in frequency spacean oracle that prepares the ground state (QAA, VQE, Spectral Combing, . . . )an oracle for time evolution (Berry et al. (2015),Hao Low et al. (2016))an oracle that prepares |E〉 = O(q)|0〉 (Roggero & Carlson (in prep.))
By performing quantum phase estimation (Kitaev(1996), Abrams&Lloyd(1999))with M ancilla qubits we will measure frequency ν with probability:
P (ν) =∑f
|〈f |E〉|2 δM (ν − Ef + E0)
finite width approximation of R(q, ω)
need only M ∼ log2 (1/∆ω) ancillaeevolution time t ∼ Poly(sys.size)/∆ω
Roggero & Carlson (in prep.)
101
102
103
104
105
106
107
108
Number of iterations
10-5
10-4
10-3
10-2
10-1
100
max
imu
m e
rro
rM = 8
M = 12
~N-0.5
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 2 / 3
Exclusive response for neutrino oscillation experiments
Goals for ν oscillation exp.neutrino massesaccurate mixing anglesCP violating phase
P (να → να) = 1− sin2(2θ)sin2(
∆m2L
4Eν
)need to use measured reaction products to constrain Eν of the event
after measuring energy ω with QPE, state-register is left in
|out〉ω ∼∑f
〈f |O(q)|0〉|f〉 with Ef − E0 = ω ±∆ω
STAY TUNED more details coming out soon: Roggero & Carlson (in prep.)
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
Exclusive response for neutrino oscillation experiments
Goals for ν oscillation exp.neutrino massesaccurate mixing anglesCP violating phase
P (να → να) = 1− sin2(2θ)sin2(
∆m2L
4Eν
)need to use measured reaction products to constrain Eν of the event
after measuring energy ω with QPE, state-register is left in
|out〉ω ∼∑f
〈f |O(q)|0〉|f〉 with Ef − E0 = ω ±∆ω
STAY TUNED more details coming out soon: Roggero & Carlson (in prep.)
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
Exclusive response for neutrino oscillation experiments
Goals for ν oscillation exp.neutrino massesaccurate mixing anglesCP violating phase
P (να → να) = 1− sin2(2θ)sin2(
∆m2L
4Eν
)need to use measured reaction products to constrain Eν of the event
after measuring energy ω with QPE, state-register is left in
|out〉ω ∼∑f
〈f |O(q)|0〉|f〉 with Ef − E0 = ω ±∆ω
STAY TUNED more details coming out soon: Roggero & Carlson (in prep.)
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
Response functions on classical computersBacca et al. (2013) LIT+CC
Lovato et al. (2016) GFMC
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
Quantum Phase Estimation
Kitaev (1996), Brassard et al. (2002), Svore et. al (2013), Weibe & Granade (2016)
QPE is a general algorithm to estimate eigenvalues of a unitary operator
U |ξk〉 = λk|ξk〉 , λk = e2πiφk ⇐ U = e−itH
starting vector |ψ〉 =∑
k ck|ξk〉store time evolution |ψ(t)〉 inauxiliary register of m qubitsperform (Quantum) Fouriertransform on the auxiliary registermeasures will return λk withprobability P (λk) ≈ |ck|2
Ovrum&Hjorth-Jensen (2007)
to get |GS〉 a good |ψ〉 is critical
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
Test on classical computer
0 0.2 0.4 0.6 0.8 1rescaled energy ω
10-5
10-4
10-3
10-2
10-1
R(ω
)
W = 8W = 12
102
104
106
108
# of iterations
10-5
10-4
10-3
10-2
10-1
100
max
err
or
2N bound state on 322 lattice
~ N-0.5
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
Final state properties from a Quantum Computerafter measuting energy ν with QPE, state-register is left in
|out〉ν ∼∑f
〈f |O(q)|0〉|f〉 with Ef − E0 = ν ±∆ω
we can then measure eg. 1- and 2-particle momentum distributions
Caveatneed to furthertime-evolve toextract informationon asymptotic statesin the detectors
STAY TUNED more details coming out soon: Roggero & Carlson (in prep.)
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
dsffsfdsfdfsfsdf
Final state properties from a Quantum Computerafter measuting energy ν with QPE, state-register is left in
|out〉ν ∼∑f
〈f |O(q)|0〉|f〉 with Ef − E0 = ν ±∆ω
we can then measure eg. 1- and 2-particle momentum distributions
Caveatneed to furthertime-evolve toextract informationon asymptotic statesin the detectors
STAY TUNED more details coming out soon: Roggero & Carlson (in prep.)
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
dsffsfdsfdfsfsdf
Final state properties from a Quantum Computerafter measuting energy ν with QPE, state-register is left in
|out〉ν ∼∑f
〈f |O(q)|0〉|f〉 with Ef − E0 = ν ±∆ω
we can then measure eg. 1- and 2-particle momentum distributions
Caveatneed to furthertime-evolve toextract informationon asymptotic statesin the detectors
STAY TUNED more details coming out soon: Roggero & Carlson (in prep.)
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
dsffsfdsfdfsfsdf
Final state properties from a Quantum Computerafter measuting energy ν with QPE, state-register is left in
|out〉ν ∼∑f
〈f |O(q)|0〉|f〉 with Ef − E0 = ν ±∆ω
we can then measure eg. 1- and 2-particle momentum distributions
Caveatneed to furthertime-evolve toextract informationon asymptotic statesin the detectors
STAY TUNED more details coming out soon: Roggero & Carlson (in prep.)
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
dsffsfdsfdfsfsdf
Final state properties from a Quantum Computerafter measuting energy ν with QPE, state-register is left in
|out〉ν ∼∑f
〈f |O(q)|0〉|f〉 with Ef − E0 = ν ±∆ω
we can then measure eg. 1- and 2-particle momentum distributions
Caveatneed to furthertime-evolve toextract informationon asymptotic statesin the detectors
STAY TUNED more details coming out soon: Roggero & Carlson (in prep.)
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
dsffsfdsfdfsfsdf
Final state properties from a Quantum Computerafter measuting energy ν with QPE, state-register is left in
|out〉ν ∼∑f
〈f |O(q)|0〉|f〉 with Ef − E0 = ν ±∆ω
we can then measure eg. 1- and 2-particle momentum distributions
Caveatneed to furthertime-evolve toextract informationon asymptotic statesin the detectors
STAY TUNED more details coming out soon: Roggero & Carlson (in prep.)
Roggero & Carlson Linear response on a QC INT - 15 Feb, 2018 3 / 3
dsffsfdsfdfsfsdf
top related