Towards a quantum LINPACK benchmarklinlin/presentations/202007...1 Towards a quantum LINPACK...
Transcript of Towards a quantum LINPACK benchmarklinlin/presentations/202007...1 Towards a quantum LINPACK...
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Towards a quantum LINPACK benchmark
Lin Lin
Department of Mathematics, UC BerkeleyLawrence Berkeley National Laboratory
Joint work with Yulong Dong (Berkeley)
Google Summer Symposium 2020,
arXiv: 2006.04010https://github.com/qsppack/racbem
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How many quantum computers will we need?Finally we have a few quantum computers
I think there is a world market for maybe five computers.–Thomas Watson, president of IBM, 1943
In 2019, there are around 2 billion computers in the world(estimate).
Prediction is very difficult, especially if it’s about the future.–Niels Bohr (also attributed to others)
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Let us make some prediction: there will be say, 10000quantum computers
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How to select the TOP500 quantum computers?
First, how to do the job for classical supercomputers?
What is LINPACK? Why LINPACK?
1https://www.top500.org/, 55th edition of the TOP500, June 2020
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LINPACK benchmark• Interested in using supercomputers for scientific computing
(instead of e.g. recognizing cats, but maybe now it is difficult todistinguish the two..)
• Solving linear systems: building block for numerous scientificcomputing applications
• LINPACK: Solve Ax = b with dense, random matrices. Noobvious applications. Supremacy?
• Controversial over its effectiveness since early days. Alternativebenchmarks have been proposed1. Still solving linear systemswith some random sparsity patterns.
• Has been used to define TOP500 since its debut in 1993.
• Quantum LINPACK benchmark?1Why Linpack no longer works as well as it once did. (link)
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Climbing the Quantum Mount Everest
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Quantum linear system problem (QLSP)
• Use quantum computers to solve
|x〉 ∝ A−1 |b〉 .
• How to get the information in A, |b〉 into a quantum computer?read-in problem.
• κ: condition number of A = ‖A‖∥∥A−1
∥∥; ε: target accuracy.
• Proper assumptions on A (e.g. d-sparse) so that oracles costpoly(n), while A ∈ C2n×2n
. (Potentially) exponential speedup.
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Compare the complexities of QLSP solversSignificant progress in the past few years: Near-optimal complexitymatching lower bounds.
Algorithm Query complexity RemarkHHL,(Harrow-Hassidim-Lloyd,2009)
O(κ2/ε) w. VTAA, complexity becomesO(κ/ε3) (Ambainis 2010)
Linear combination of unitaries(LCU),(Childs-Kothari-Somma,2017)
O(κ2polylog(1/ε)) w. VTAA, complexity becomesO(κ poly log(1/ε))
Quantum singular value transfor-mation (QSVT) (Gilyén-Su-Low-Wiebe, 2019)
O(κ2 log(1/ε)) Queries the RHS only O(κ) times
Randomization method (RM)(Subasi-Somma-Orsucci, 2019)
O(κ/ε) Prepares a mixed state; w. re-peated phase estimation, complex-ity becomes O(κ poly log(1/ε))
Time-optimal adiabatic quantumcomputing (AQC(exp)) (An-L.,2019, 1909.05500)
O(κ poly log(1/ε)) No need for any amplitude amplifi-cation. Use time-dependent Hamil-tonian simulation.
Eigenstate filtering (L.-Tong, 2019,1910.14596)
O(κ log(1/ε)) No need for any amplitude amplifi-cation. Does not rely on any com-plex subroutines.
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Quantum benchmark problem
• All these algorithms require full fault-tolerant computers to getanywhere.
• Getting the matrix into quantum computer alone (using e.g. LCU)can be prohibitively expensive on near-term devices.
• Will likely remain so for some time for real applications.
• Quantum LINPACK benchmark is different: do we really need /want to generate random numbers classically and get them intoquantum computers say via QRAM?
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Idea: from the success of Google’s supremacy circuit
• A big random, unitary matrix, almost drawn from Haar measure.
• Linear algebra usually works with non-unitary matrices.
• How about taking the upper-left diagonal block n-qubit matrix ofthe (n + 1)-qubit unitary matrix (can be random)?
UA =
(A ·· ·
)• FACT: It can represent in principle any n-qubit matrix (up to
scaling)!
• Block-encoding (Gilyén-Su-Low-Wiebe, 2019)
• RAndom Circuit Block-Encoded Matrix (RACBEM)
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RACBEM
• A very flexible way to construct a non-unitary matrix with respectto any coupling map of the quantum architecture.
• Take upper-left diagonal block: measure one-qubit.A = (〈0| ⊗ In)UA (|0〉 ⊗ In)
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Error of RACBEM on IBM Q
1 2 3 4 5 6 7 8n_sys_qubit
0.0
0.1
0.2
0.3
0.4
0.5
0.6
|pm
eas
p exa
ct|/p
exac
t
Relative error in success probability of obtaining A |0n〉 for differentnumber of system qubits, on the 5-qubit backend ibmq_burlington,and 15-qubit backend ibmq_16_melbourne.
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Representing matrix functions, say f (A) = A−1
• General quantum singular value transformation circuit (QSVT)(Gilyén-Su-Low-Wiebe, 2019). Always even order polynomial,and 1 extra ancilla qubit.
• Follow the natural layout of the quantum circuit. Can run withouteven calling a transpiler.
• Applications: Linear systems, time series, spectral measure,thermal energy ... with Hermitian-RACBEM.
• How to obtain the phase factors: optimization based approach toget > 10000 phase factors 1
1(Dong-Meng-Whaley-L., 2002.11649). https://github.com/qsppack/qsppack
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Solving linear system on IBM Q and QVM
Compute∥∥H−1 |0n〉
∥∥22. (sigma: noise level on QVM)
Well conditioned linear system
ibmq_london ibmq_burlington ibmq_essex ibmq_ourense ibmq_vigoIBM Q backend (5-qubit)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
|pm
eas
p exa
ct|/p
exac
t
length of QSVTphase factors
311
0.0 0.2 0.4 0.6 0.8 1.0sigma
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
|pm
eas
p exa
ct|/p
exac
t
(a) = 2
0.0 0.2 0.4 0.6 0.8 1.0sigma
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
0.225
|pm
eas
p exa
ct|/p
exac
t
(b) = 5
0.0 0.2 0.4 0.6 0.8 1.0sigma
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
|pm
eas
p exa
ct|/p
exac
t
(c) = 10
0.0 0.2 0.4 0.6 0.8 1.0sigma
0.2
0.4
0.6
0.8
1.0
|pm
eas
p exa
ct|/p
exac
t
(d) = 20
n_sys_qubit 3 4 5 6 7 8 9 10
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Time series (no Trotter)
s(t) = 〈ψ|eiHt |ψ〉
QVM:
2 4 6 8 10
0.50
0.25
0.00
0.25
0.50
0.75
Re0n |e
iHt |0
n
(i) n_sys_qubit = 7
2 4 6 8 10
0.25
0.00
0.25
0.50
0.75Re
0n |eiH
t |0n
(ii) n_sys_qubit = 8
2 4 6 8 10t
0.2
0.0
0.2
0.4
0.6
Im0n |e
iHt |0
n
2 4 6 8 10t
0.2
0.0
0.2
0.4
0.6
Im0n |e
iHt |0
n
2 4 6 8 10
0.5
0.0
0.5
Re0n |e
iHt |0
n
(iii) n_sys_qubit = 9
2 4 6 8 10
0.50
0.25
0.00
0.25
0.50
0.75
Re0n |e
iHt |0
n
(iv) n_sys_qubit = 10
2 4 6 8 10t
0.50
0.25
0.00
0.25
0.50
Im0n |e
iHt |0
n
2 4 6 8 10t
0.4
0.2
0.0
0.2
0.4
0.6
Im0n |e
iHt |0
n
exact, realexact, imag
QSVT without error, realQSVT without error, imag
sigma = 0.00, realsigma = 0.00, imag
sigma = 0.25, realsigma = 0.25, imag
sigma = 0.50, realsigma = 0.50, imag
sigma = 0.75, realsigma = 0.75, imag
sigma = 1.00, realsigma = 1.00, imag
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Spectral measure
s(E) = 〈ψ|δ(H− E)|ψ〉 ≈ 1πIm 〈ψ|(H− E − iη)−1|ψ〉
QVM:
0.00 0.25 0.50 0.75 1.00E
0.4
0.6
0.8
1.0
1.2sp
ectra
l mea
sure
(a) n_sys_qubit = 7
0.00 0.25 0.50 0.75 1.00E
0.4
0.6
0.8
1.0
1.2
1.4
spec
tral m
easu
re
(b) n_sys_qubit = 8
0.00 0.25 0.50 0.75 1.00E
0.4
0.5
0.6
0.7
0.8
0.9
spec
tral m
easu
re
(c) n_sys_qubit = 9
0.00 0.25 0.50 0.75 1.00E
0.4
0.6
0.8
1.0sp
ectra
l mea
sure
(d) n_sys_qubit = 10
exactQSVT without noise
sigma = 0.00sigma = 0.25
sigma = 0.50sigma = 0.75
sigma = 1.00
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Thermal energy
E(β) =Tr[He−βH]Tr[e−βH]
IBM Q (left) and QVM (right)
2 4 6 8
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
E()
(i) n_sys_qubit = 1
2 4 6 8
0.1
0.2
0.3
0.4
0.5E(
)(ii) n_sys_qubit = 2
2 4 6 80.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
E()
(iii) n_sys_qubit = 3
2 4 6 8
0.1
0.2
0.3
0.4
E()
(iv) n_sys_qubit = 5
exact IBM Q
2 4 6 80.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
E()
(i) n_sys_qubit = 3backend = ibmq_essex (5-qubit)
2 4 6 8
0.1
0.2
0.3
0.4
E()
(ii) n_sys_qubit = 3backend = ibmq_16_melbourne (15-qubit)
2 4 6 80.05
0.10
0.15
0.20
0.25
0.30
0.35
E()
(iii) n_sys_qubit = 5backend = ibmq_16_melbourne (15-qubit)
2 4 6 8
0.10
0.15
0.20
0.25
0.30
0.35
0.40
E()
(iv) n_sys_qubit = 7backend = ibmq_16_melbourne (15-qubit)
exactsigma = 0.00
sigma = 0.25sigma = 0.50
sigma = 0.75sigma = 1.00
1Use the minimally entangled typical thermal state (METTS) algorithm (White,2009) (Motta et al, 2020)
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Conclusion
• Linear system with (Hermitian-)RACBEM: Quantum LINPACKbenchmark
• Uses a supremacy circuit as building block.
• Can be easily engineered w.r.t. almost any architecture.
• Maybe only steps away from the supremacy test.
• Hardness? Each UA is already hard..
• More quantitative ways to measure success and classicalhardness: cross-entropy test etc.
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References
https://github.com/qsppack/qsppackhttps://github.com/qsppack/racbem• Y. Dong and L. Lin, Random circuit block-encoded matrix and a
proposal of quantum LINPACK benchmark [arXiv:2006.04010]
• Y. Dong, X. Meng, K. B. Whaley, L. Lin, Efficient Phase FactorEvaluation in Quantum Signal Processing [arXiv:2002.11649]
• L. Lin and Y. Tong, Optimal quantum eigenstate filtering withapplication to solving quantum linear systems [arXiv:1910.14596]
• D. An and L. Lin, Quantum linear system solver based ontime-optimal adiabatic quantum computing and quantumapproximate optimization algorithm [arXiv:1909.05500]
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Acknowledgement
Thank you for your attention!
Lin Linhttps://math.berkeley.edu/~linlin/
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Two issues of RACBEM
• Not Hermitian.
• Matrix becomes increasingly singular as n increases.
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Fixing both issues: Hermitian-RACBEM
• Simple quantum singular value transformation circuit (QSVT)(Gilyén-Su-Low-Wiebe, 2019) of degree 2.
• Explicit formulation of the Hermitian matrix by tracing out 2ancilla (the extra ancilla can be reused later).
H = [−2 sin(2ϕ0) sinϕ1]A†A + cos(2ϕ0 − ϕ1).
• Fully adjustable condition number.
• This is the really useful thing.