cs269q.stanford.edu · 2019-07-31 · 19mm Aluminum circuit on Silicon Circuit QED: Blais et al,...
Transcript of cs269q.stanford.edu · 2019-07-31 · 19mm Aluminum circuit on Silicon Circuit QED: Blais et al,...
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Quantum Approximate Optimization Algorithm[QAOA] Hybrid algorithm used for constraint satisfaction problems
pronounced : kwaah-waah
Given binary constraints: MAXIMIZE
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“Maximize disagreement on a colored graph”
Score 0 Score 0 Score+1
The MaxCut problem
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“Maximize disagreement on a colored graph”
Score 0 Score 0 Score+1
Score = +8 (max)
8-node “ring of disagrees”
The MaxCut problem
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“Maximize disagreement on a colored graph”
Score 0 Score 0 Score+1
MaxCut on a quantum computer
Score+1
Score 0Score = +8 (max)
8-node “ring of disagrees”
The MaxCut problem
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Solving MAXCUT with QAOA
MAXCUT is an NP-complete problem
A quantum solver has at most a polynomial advantage for exact solution.
However, the Quantum Approximate Optimization Algorithm (QAOA) [Fahri et al, 2014] is a heuristic approach that has been shown to be competitive with the best classical algorithms.
There is a form of supremacy as well [Farhi & Harrow 2016]
QAOA: Farhi, Goldstone, and Gutman arXiv: 1411.4028 (2014) MAXCUT & QAOA: Z. Wang et al, arXiv: 1706.02998 (2017)
Farhi & Harrow arXiv: 602.07674 (2016)
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The Quantum Approximate Optimization Algorithm
Inspired by adiabatic quantum computing
“Cost” Hamiltonian
“Driver” Hamiltonian
Farhi et. al., 2014
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The Quantum Approximate Optimization Algorithm
Inspired by adiabatic quantum computing
Discretize evolution into P steps
“Cost” Hamiltonian
“Driver” Hamiltonian
Farhi et. al., 2014
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The Quantum Approximate Optimization Algorithm
Inspired by adiabatic quantum computing
Discretize evolution into P steps
P successive applications of Cost and Driver Unitaries
“Cost” Hamiltonian
“Driver” Hamiltonian
Farhi et. al., 2014
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QAOA
0. Prepare the initial state
The procedure
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1. Apply the cost Hamiltonian
2. Apply the driver Hamiltonian
QAOA
0. Prepare the initial state
The procedure
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1. Apply the cost Hamiltonian
2. Apply the driver Hamiltonian
QAOA
3. Exponentiate, parameterize in P steps by P Betas and P Gammas
0. Prepare the initial state
The procedure
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1. Apply the cost Hamiltonian
2. Apply the driver Hamiltonian
QAOA
3. Exponentiate, parameterize in P steps by P Betas and P Gammas
0. Prepare the initial state
The procedure
4. Optimize over betas and gammas
QPU/QVM(QAOA ansatz)
CPU(Bayesian
Optimization)
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Staff Scheduling Problem (NP-complete)See black board notes
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k-means clustering
> Given an unlabeled set of points
Otterbach, … WZ, ... et al 1712.05771 Unsupervised Machine Learning on a Hybrid Quantum Computer
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k-means clustering
> Given an unlabeled set of points,
> find labels based upon similarity metric (e.g. Euclidean distance).
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“Maximize disagreement on a colored graph”
Score 0 Score 0 Score+1
MaxCut on a quantum computer
Score+1
Score 0Score = +8 (max)
8-node “ring of disagrees”
The MaxCut problem
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“Maximize disagreement on a colored graph”
Score 0 Score 0 Score+𝛂
MaxCut on a quantum computer
Score+1
Score 0Score = sum(𝛂ij)
8-node “ring of disagrees”
The weighted MaxCut problem
𝛂 𝛂 𝛂
* 𝛂ij
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Construct a graph G=(V,E) where the edge weights w_i,j are determined by the distance metric.
Then, MAXCUT is a clustering algorithm for the original points.
2-means clustering as MAXCUT
MAXCUT =
2
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Construct a graph G=(V,E) where the edge weights w_i,j are determined by the distance metric.
Then, MAXCUT is a clustering algorithm for the original points.
Clustering transformed into an optimization problem.
2-means clustering as MAXCUT
MAXCUT =
2
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19mm
Aluminum circuit on Silicon
Circuit QED:Blais et al, PRA 69, 062320 (2004)
Wallraff et al, Nature 431, 162 (2004)Hutchings et al, quant-ph/1702.02253 (2017)
Rigetti 19Q
Device Properties● 4x5 lattice of transmon qubits and
quasi-lumped element resonators
● Fixed capacitive coupling between qubits
● Alternating arrangement of fixed-frequency and tunable (asymmetric) transmon qubits
● “19Q” because one tunable qubit was not tunable
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19mm
Aluminum circuit on Silicon
Circuit QED:Blais et al, PRA 69, 062320 (2004)
Wallraff et al, Nature 431, 162 (2004)Hutchings et al, quant-ph/1702.02253 (2017)
Rigetti 19Q
Device Properties● 4x5 lattice of transmon qubits and
quasi-lumped element resonators
● Fixed capacitive coupling between qubits
● Alternating arrangement of fixed-frequency and tunable (asymmetric) transmon qubits
● T1 = 8-30 μs, T2* = 5-25 μs
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19Q connectivity graph
Inspired by:FM gate theory: Beaudoin et al PRA 86, 022305 (2012) experiment: Strand et al, PRB 87, 220505(R) (2013)
B-tune gate: McKay et al Phys Rev Applied 6, 064007 (2016)
Qubit-qubit interactions
● Fixed coupling between fixed-frequency and tunable (asymmetric) transmon qubits
● 2-qubit parametric gates use RF flux modulation to turn on effective resonance conditions
● Typical 2-qubit gate fidelity of 0.85-0.95
Parametric gates:theory: N. Didier et al, arXiv:1706.06566 (2017)experiment: S. Caldwell et al, arXiv:1706.06562 (2017)
Rigetti 19Q
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19Q connectivity graph
Inspired by:FM gate theory: Beaudoin et al PRA 86, 022305 (2012) experiment: Strand et al, PRB 87, 220505(R) (2013)
B-tune gate: McKay et al Phys Rev Applied 6, 064007 (2016)
Qubit-qubit interactions
● Fixed coupling between fixed-frequency and tunable (asymmetric) transmon qubits
● 2-qubit parametric gates use RF flux modulation to turn on effective resonance conditions
● Typical 2-qubit gate fidelity of 0.85-0.95
Parametric gates:theory: N. Didier et al, arXiv:1706.06566 (2017)experiment: S. Caldwell et al, arXiv:1706.06562 (2017)
Rigetti 19Q
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Sparse similarity metrics
Similarity is measured by a distance metric over the feature vector. In some domains, similarity can be measured by an overlap metric, leading to sparse graphs.
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Sparse graphs with 19Q connectivity
Generate a family of sparse graphs with random weights matching the connectivity of 19Q.
This allows implementation of HC in a circuit of depth 3 (becomes depth 6 after compilation)
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Clustering on 19Q
We run QAOA with p=1on 19Q. The average cost is typically quite low, but we observe some samples close to the optimal solution.
We use a Bayesian approach to choose β, ɣ
QAOA with 2,500 samples at each step
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Clustering on 19Q83 trials for a fixed problem instance
In many such trials, the algorithm actually finds the optimal solution.
From these trials we calculate an empirical CDF.
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Clustering performance
Success probability monotonically increases with number of steps.
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Clustering performance
Success probability monotonically increases with number of steps.
Noise in 19Q has a significant impact on performance.
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Clustering performance
Success probability monotonically increases with number of steps.
Noise in 19Q has a significant impact on performance.
Approach clearly outperforms random sampling.
Otterbach, … WZ, ... et al 1712.05771 Unsupervised Machine Learning on a Hybrid Quantum Computer