Quantum Algorithm Survey: Cont d
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1 Quantum Algorithm Survey: Cont d Xiaodi Wu CS, UMIACS, and QuICS, University of Maryland
2 Quantum Algorithms for Linear Algebraic Problems Linear equation systems A x = b, given oracle access to A, b, generate x x Ref. [HHL]: the first quantum algorithm (poly-log dimension dependence) based on phase-estimation. Ref. [CKS]: poly-log error dependence based on approximation theory, linear combination of unitaries (LCU), quantum walks and Hamiltonian simulation.
3 Quantum Algorithms for Linear Algebraic Problems Linear equation systems A x = b, given oracle access to A, b, generate x x Ref. [HHL]: the first quantum algorithm (poly-log dimension dependence) based on phase-estimation. Ref. [CKS]: poly-log error dependence based on approximation theory, linear combination of unitaries (LCU), quantum walks and Hamiltonian simulation. More examples Singular value decomposition, matrix factorization,...
4 Quantum Algorithms for Linear Algebraic Problems Linear equation systems A x = b, given oracle access to A, b, generate x x Ref. [HHL]: the first quantum algorithm (poly-log dimension dependence) based on phase-estimation. Ref. [CKS]: poly-log error dependence based on approximation theory, linear combination of unitaries (LCU), quantum walks and Hamiltonian simulation. More examples Singular value decomposition, matrix factorization,... Spectral method, linear sketches,...
5 Quantum Algorithms for Machine Learning Many proposals Quantum algorithms for supper-vector machines, principle component analysis,...
6 Quantum Algorithms for Machine Learning Many proposals Quantum algorithms for supper-vector machines, principle component analysis,... Quantum algorithms for data fitting, geometric analysis of data, linear regression...
7 Quantum Algorithms for Machine Learning Many proposals Quantum algorithms for supper-vector machines, principle component analysis,... Quantum algorithms for data fitting, geometric analysis of data, linear regression... Quantum enhanced supervised and unsupervised learning, reinforcement learning, deep learning...
8 Quantum Algorithms for Machine Learning Many proposals Quantum algorithms for supper-vector machines, principle component analysis,... Quantum algorithms for data fitting, geometric analysis of data, linear regression... Quantum enhanced supervised and unsupervised learning, reinforcement learning, deep learning... Quantum learning of quantum systems,...
9 Quantum Algorithms for Machine Learning Many proposals Quantum algorithms for supper-vector machines, principle component analysis,... Quantum algorithms for data fitting, geometric analysis of data, linear regression... Quantum enhanced supervised and unsupervised learning, reinforcement learning, deep learning... Quantum learning of quantum systems,...
10 Quantum Algorithms for Machine Learning Many proposals Quantum algorithms for supper-vector machines, principle component analysis,... Quantum algorithms for data fitting, geometric analysis of data, linear regression... Quantum enhanced supervised and unsupervised learning, reinforcement learning, deep learning... Quantum learning of quantum systems,... Also note that: No much quantum speed-up in the sense of computational learning theory.
11 Quantum Algorithms for Optimizations Optimizations Ref. [BS, vaggdw, BKL+] Quadratic improvement over classical algorithms.
12 Quantum Algorithms for Optimizations Optimizations Ref. [BS, vaggdw, BKL+] Quadratic improvement over classical algorithms. Poly-log dependence on the dimension with quantum inputs.
13 Quantum Algorithms for Optimizations Optimizations Ref. [BS, vaggdw, BKL+] Quadratic improvement over classical algorithms. Poly-log dependence on the dimension with quantum inputs. Quantum acceleration of the gradient descent method.
14 Quantum Algorithms for Optimizations Optimizations Ref. [BS, vaggdw, BKL+] Quadratic improvement over classical algorithms. Poly-log dependence on the dimension with quantum inputs. Quantum acceleration of the gradient descent method.
15 Quantum Algorithms for Optimizations Optimizations Ref. [BS, vaggdw, BKL+] Quadratic improvement over classical algorithms. Poly-log dependence on the dimension with quantum inputs. Quantum acceleration of the gradient descent method. Open questions Concrete examples of SDP with provable quantum speed-up. Optimization beyond SDPs. Quantize other classical techniques.
16 Variational Methods based on Short-depth Circuits ,0 2 1, ,0 or 1 Y 1 1,0 qubit 1 1,0 ( ) 1,1 ( ) 1, D( ) qubit 2 2,0 ( ) 2,1 ( ) 2, D( ) ent ent qubit,0 ( ),1 ( ), D( ) 0 ( ) 1 ( ) D( ) D times Figure 4: Heuristic preparation of trial states for the variational quantum eigensolver based on single-qubit gates U( ) interleavedbyentanglingoperationsu ent as described in the text. Independently of the particular problem to be solved, one may choose trial states that can be e ciently generated in current quantum hardware and at the same time allow the generation of highly entangled states that are close to the target state. This approach is showcased in the examples provided in Sections 4.4 and 5.1. As shown in Fig. 4, thepreparationoftheheuristictrialstatescomprisestwotypesof quantum gates, single-qubit Euler rotations U( ) determinedbytherotationangles and an entangling drift operation U ent acting on pairs of qubits. The N-qubit trial states are obtained by applying a sequence of D entanglers U ent alternating with the Euler rotations on the N qubits to the initial ground state i, D times z } { ( )i = U D ( )U ent...u 1 ( )U ent U 0 ( ) i (13)
17 Variational Methods based on Short-depth Circuits ,0 2 1, ,0 or 1 Y 1 1,0 qubit 1 1,0 ( ) 1,1 ( ) 1, D( ) qubit 2 2,0 ( ) 2,1 ( ) 2, D( ) ent ent qubit,0 ( ),1 ( ), D( ) 0 ( ) 1 ( ) D( ) D times Proposals Figure 4: Heuristic preparation of trial states for the variational quantum eigensolver based on single-qubit gates U( ) interleavedbyentanglingoperationsu ent as described in the text. Quantum chemistry: ground energy of Hamiltonians. Independently of the particular problem to be solved, one may choose trial states that can be e ciently generated in current quantum hardware and at the same time allow the generation of highly entangled states that are close to the target state. This approach is showcased in the examples provided in Sections 4.4 and 5.1. As shown in Fig. 4, thepreparationoftheheuristictrialstatescomprisestwotypesof quantum gates, single-qubit Euler rotations U( ) determinedbytherotationangles and an entangling drift operation U ent acting on pairs of qubits. The N-qubit trial states are obtained by applying a sequence of D entanglers U ent alternating with the Euler rotations on the N qubits to the initial ground state i, D times z } { ( )i = U D ( )U ent...u 1 ( )U ent U 0 ( ) i (13)
18 Variational Methods based on Short-depth Circuits ,0 2 1, ,0 or 1 Y 1 1,0 qubit 1 1,0 ( ) 1,1 ( ) 1, D( ) qubit 2 2,0 ( ) 2,1 ( ) 2, D( ) ent ent qubit,0 ( ),1 ( ), D( ) 0 ( ) 1 ( ) D( ) D times Proposals Figure 4: Heuristic preparation of trial states for the variational quantum eigensolver based on single-qubit gates U( ) interleavedbyentanglingoperationsu ent as described in the text. Quantum chemistry: ground energy of Hamiltonians. Independently of the particular problem to be solved, one may choose trial states that can be e ciently generated in current quantum hardware and at the same time allow the generation of highly entangled states that are close to the target state. This approach is showcased in the examples provided in Sections 4.4 and 5.1. As shown in Fig. 4, thepreparationoftheheuristictrialstatescomprisestwotypesof quantum gates, single-qubit Euler rotations U( ) determinedbytherotationangles and an entangling drift operation U ent acting on pairs of qubits. The N-qubit trial states are obtained by applying a sequence of D entanglers U ent alternating with the Euler rotations on the N qubits to the initial ground state i, Optimization: Quantum Approximate Optimization Algorithm (QAOA) D times z } { ( )i = U D ( )U ent...u 1 ( )U ent U 0 ( ) i (13)
19 Variational Methods based on Short-depth Circuits ,0 2 1, ,0 or 1 Y 1 1,0 qubit 1 1,0 ( ) 1,1 ( ) 1, D( ) qubit 2 2,0 ( ) 2,1 ( ) 2, D( ) ent ent qubit,0 ( ),1 ( ), D( ) 0 ( ) 1 ( ) D( ) D times Proposals Figure 4: Heuristic preparation of trial states for the variational quantum eigensolver based on single-qubit gates U( ) interleavedbyentanglingoperationsu ent as described in the text. Quantum chemistry: ground energy of Hamiltonians. Independently of the particular problem to be solved, one may choose trial states that can be e ciently generated in current quantum hardware and at the same time allow the generation of highly entangled states that are close to the target state. This approach is showcased in the examples provided in Sections 4.4 and 5.1. As shown in Fig. 4, thepreparationoftheheuristictrialstatescomprisestwotypesof quantum gates, single-qubit Euler rotations U( ) determinedbytherotationangles and an entangling drift operation U ent acting on pairs of qubits. The N-qubit trial states are obtained by applying a sequence of D entanglers U ent alternating with the Euler rotations on the N qubits to the initial ground state i, Optimization: Quantum Approximate Optimization Algorithm (QAOA) Quantum Neural Network. D times z } { ( )i = U D ( )U ent...u 1 ( )U ent U 0 ( ) i (13)
20 Variational Methods based on Short-depth Circuits ,0 2 1, ,0 or 1 Y 1 1,0 qubit 1 1,0 ( ) 1,1 ( ) 1, D( ) qubit 2 2,0 ( ) 2,1 ( ) 2, D( ) ent ent qubit,0 ( ),1 ( ), D( ) 0 ( ) 1 ( ) D( ) D times Proposals Figure 4: Heuristic preparation of trial states for the variational quantum eigensolver based on single-qubit gates U( ) interleavedbyentanglingoperationsu ent as described in the text. Quantum chemistry: ground energy of Hamiltonians. Independently of the particular problem to be solved, one may choose trial states that can be e ciently generated in current quantum hardware and at the same time allow the generation of highly entangled states that are close to the target state. This approach is showcased in the examples provided in Sections 4.4 and 5.1. As shown in Fig. 4, thepreparationoftheheuristictrialstatescomprisestwotypesof quantum gates, single-qubit Euler rotations U( ) determinedbytherotationangles and an entangling drift operation U ent acting on pairs of qubits. The N-qubit trial states are obtained by applying a sequence of D entanglers U ent alternating with the Euler rotations on the N qubits to the initial ground state i, Optimization: Quantum Approximate Optimization Algorithm (QAOA) Quantum Neural Network. Representing kernels, e.g., applied in, support vector D times machines (SVM) z } { ( )i = U D ( )U ent...u 1 ( )U ent U 0 ( ) i (13)
21 Classical-Quantumfermionic Hybrid problem Algorithms qubit Hamiltonian classical cost function calculate energy prepare trial state classical adjust parameters optimize measure expectation values quantum solution Figure 3: Variational quantum eigensolver method. The trial states, which depend on a few classical parameters, arecreatedonthequantumdeviceandusedformeasuring the expectation values needed. These are combined on a classical computer to calculate the energy E q( ), i.e. the cost function, and find new parameters to minimize it. The new parameters are then fed back into the algorithm. The parameters of the solution are obtained when the minimal energy is reached. qubit quantum states ( )i parametrized by control parameters. The subsequent measurement of a cost function E q( ) = h ( ) H q ( )i, typically the energy of a problem Hamiltonian H q,servesaclassicalcomputertofindnewvalues in order to minimize E q( ) andfindtheground-stateenergy E min q =min (h ( ) H q ( )i). (3) This variational quantum eigensolver approach to Hamiltonian-problem solving has been
22 Classical-Quantumfermionic Hybrid problem Algorithms qubit Hamiltonian classical cost function calculate energy prepare trial state classical adjust parameters optimize measure expectation values quantum solution Figure 3: Variational quantum eigensolver method. The trial states, which depend on a few classical parameters, arecreatedonthequantumdeviceandusedformeasuring the expectation values needed. These are combined on a classical computer to calculate the energy E q( ), i.e. the cost function, and find new parameters to minimize it. The new parameters are then fed back into the algorithm. The parameters of the solution are obtained when the minimal energy is reached. Reasonable scenario for algorithm design on near-term quantum devices. qubit quantum states ( )i parametrized by control parameters. The subsequent measurement of a cost function E q( ) = h ( ) H q ( )i, typically the energy of a problem Hamiltonian H q,servesaclassicalcomputertofindnewvalues in order to minimize E q( ) andfindtheground-stateenergy Eq min =min(h ( ) H q ( )i). (3) This variational quantum eigensolver approach to Hamiltonian-problem solving has been
23 Classical-Quantumfermionic Hybrid problem Algorithms qubit Hamiltonian classical cost function calculate energy prepare trial state classical adjust parameters optimize measure expectation values quantum solution Figure 3: Variational quantum eigensolver method. The trial states, which depend on a few classical parameters, arecreatedonthequantumdeviceandusedformeasuring the expectation values needed. These are combined on a classical computer to calculate the energy E q( ), i.e. the cost function, and find new parameters to minimize it. The new parameters are then fed back into the algorithm. The parameters of the solution are obtained when the minimal energy is reached. Reasonable scenario for algorithm design on near-term quantum devices. Heterogeneous computing, architecture, system, programming languages. qubit quantum states ( )i parametrized by control parameters. The subsequent measurement of a cost function E q( ) = h ( ) H q ( )i, typically the energy of a problem Hamiltonian H q,servesaclassicalcomputertofindnewvalues in order to minimize E q( ) andfindtheground-stateenergy Eq min =min(h ( ) H q ( )i). (3) This variational quantum eigensolver approach to Hamiltonian-problem solving has been
24 Thank You!! Q & A
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