Quantum Algorithms for Estimating Physical Quantities using - PowerPoint PPT Presentation

Quantum Algorithms for Estimating Physical Quantities using Block-Encodings Patrick Rall Quantum Information Center University of Texas at Austin A review of modern techniques and new results from arXiv:2004.06832 May 2020 Patrick Rall Algorithms from Block Encodings May 2020 1 / 22

Quantum Primitives | 0 � U | 0 � Initialization Patrick Rall Algorithms from Block Encodings May 2020 2 / 22

Quantum Primitives | 0 � U | 0 � Initialization Unitary evolution Patrick Rall Algorithms from Block Encodings May 2020 2 / 22

Quantum Primitives | 0 � U | 0 � Initialization Unitary evolution Measurement Patrick Rall Algorithms from Block Encodings May 2020 2 / 22

Quantum Primitives | 0 � | 0 � U | 0 � Initialization Unitary evolution ❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭ Measurement ❤ Postselection Patrick Rall Algorithms from Block Encodings May 2020 2 / 22

Why postselection as a primitive? Postselection captures common operations Estimate probability of postselection success Patrick Rall Algorithms from Block Encodings May 2020 3 / 22

Why postselection as a primitive? Postselection captures common operations Estimate probability of postselection success Condition experiment on postselection success Patrick Rall Algorithms from Block Encodings May 2020 3 / 22

Why postselection as a primitive? Postselection captures common operations Estimate probability of postselection success Condition experiment on postselection success Quadratic speedups for both of these common operations. Patrick Rall Algorithms from Block Encodings May 2020 3 / 22

Why postselection as a primitive? Postselection captures common operations Estimate probability of postselection success Classical: To get precision ε , need O (1 /ε 2 ) samples Condition experiment on postselection success Quadratic speedups for both of these common operations. Patrick Rall Algorithms from Block Encodings May 2020 3 / 22

Why postselection as a primitive? Postselection captures common operations Estimate probability of postselection success Classical: To get precision ε , need O (1 /ε 2 ) samples Quantum: Amplitude estimation has circuit size O (1 /ε ) Condition experiment on postselection success Quadratic speedups for both of these common operations. Patrick Rall Algorithms from Block Encodings May 2020 3 / 22

Why postselection as a primitive? Postselection captures common operations Estimate probability of postselection success Classical: To get precision ε , need O (1 /ε 2 ) samples Quantum: Amplitude estimation has circuit size O (1 /ε ) Condition experiment on postselection success Classical: If success probability is p , to try O (1 / p ) times Quadratic speedups for both of these common operations. Patrick Rall Algorithms from Block Encodings May 2020 3 / 22

Why postselection as a primitive? Postselection captures common operations Estimate probability of postselection success Classical: To get precision ε , need O (1 /ε 2 ) samples Quantum: Amplitude estimation has circuit size O (1 /ε ) Condition experiment on postselection success Classical: If success probability is p , to try O (1 / p ) times Quantum: Amplitude amplification has circuit size O (1 / √ p ) Quadratic speedups for both of these common operations. Patrick Rall Algorithms from Block Encodings May 2020 3 / 22

Example: Expectation of Pauli Matrix Some state | ψ � and Pauli matrix P . Estimate � ψ | P | ψ � Let U | 0 n � = | ψ � . Patrick Rall Algorithms from Block Encodings May 2020 4 / 22

Example: Expectation of Pauli Matrix Some state | ψ � and Pauli matrix P . Estimate � ψ | P | ψ � Let U | 0 n � = | ψ � . Standard method: Let VPV † = I ⊗ σ Z ⊗ σ Z ⊗ I Patrick Rall Algorithms from Block Encodings May 2020 4 / 22

Example: Expectation of Pauli Matrix Some state | ψ � and Pauli matrix P . Estimate � ψ | P | ψ � Let U | 0 n � = | ψ � . Standard method: Let VPV † = I ⊗ σ Z ⊗ σ Z ⊗ I | 0 � n V † U Patrick Rall Algorithms from Block Encodings May 2020 4 / 22

Example: Expectation of Pauli Matrix Some state | ψ � and Pauli matrix P . Estimate � ψ | P | ψ � Let U | 0 n � = | ψ � . Standard method: Let VPV † = I ⊗ σ Z ⊗ σ Z ⊗ I | 0 � n V † U Alternate method: exploit that P is unitary Patrick Rall Algorithms from Block Encodings May 2020 4 / 22

Example: Expectation of Pauli Matrix Some state | ψ � and Pauli matrix P . Estimate � ψ | P | ψ � Let U | 0 n � = | ψ � . Standard method: Let VPV † = I ⊗ σ Z ⊗ σ Z ⊗ I | 0 � n V † U Alternate method: exploit that P is unitary | 0 � n | 0 � U † U P Patrick Rall Algorithms from Block Encodings May 2020 4 / 22

Example: Expectation of Pauli Matrix Some state | ψ � and Pauli matrix P . Estimate � ψ | P | ψ � Let U | 0 n � = | ψ � . Standard method: Let VPV † = I ⊗ σ Z ⊗ σ Z ⊗ I | 0 � n V † U Alternate method: exploit that P is unitary | 0 � n | 0 � U † U P Postselection probability: |� ψ | P | ψ �| 2 . Almost what we want. Patrick Rall Algorithms from Block Encodings May 2020 4 / 22

Example: Expectation of Pauli Matrix Smallest eigenvalue of P is − 1, so: � � � ψ | I + P � � = � ψ | I + P | ψ � = � ψ | P | ψ � + 1 � � | ψ � � � 2 2 2 Patrick Rall Algorithms from Block Encodings May 2020 5 / 22

Example: Expectation of Pauli Matrix Smallest eigenvalue of P is − 1, so: � � ψ | I + P � � � = � ψ | I + P | ψ � = � ψ | P | ψ � + 1 � � | ψ � � � 2 2 2 If only we could multiply by I + P rather than P ... 2 Patrick Rall Algorithms from Block Encodings May 2020 5 / 22

Example: Expectation of Pauli Matrix Smallest eigenvalue of P is − 1, so: � � � ψ | I + P � � = � ψ | I + P | ψ � = � ψ | P | ψ � + 1 � � | ψ � � � 2 2 2 If only we could multiply by I + P rather than P ... 2 | + � • | + � | 0 � n U † | 0 � U P Patrick Rall Algorithms from Block Encodings May 2020 5 / 22

Example: Expectation of Pauli Matrix Smallest eigenvalue of P is − 1, so: � � � ψ | I + P � = � ψ | I + P � | ψ � = � ψ | P | ψ � + 1 � � | ψ � � � 2 2 2 If only we could multiply by I + P rather than P ... 2 | + � • | + � | 0 � n U † | 0 � U P Observe that ( � + | ⊗ I )CTRL-P( | + � ⊗ I ) = I + P 2 Patrick Rall Algorithms from Block Encodings May 2020 5 / 22

Probabilistic mixtures of unitaries � | ψ � → A | ψ � where A = p i U i i Patrick Rall Algorithms from Block Encodings May 2020 6 / 22

Probabilistic mixtures of unitaries � | ψ � → A | ψ � where A = p i U i i √ p i | i � √ p i | i � � � • i i | ψ � U i Patrick Rall Algorithms from Block Encodings May 2020 6 / 22

Probabilistic mixtures of unitaries � | ψ � → A | ψ � where A = p i U i i √ p i | i � √ p i | i � � � • i i | ψ � U i � SELECT( U ) = | i � � i | ⊗ U i i Patrick Rall Algorithms from Block Encodings May 2020 6 / 22

Probabilistic mixtures of unitaries � | ψ � → A | ψ � where A = p i U i i √ p i | i � √ p i | i � � � • i i | ψ � U i � SELECT( U ) = | i � � i | ⊗ U i i Can ‘perform’ non-unitary operations! Berry, Childs, Kothari, Somma - arXiv:1501.01715, arXiv:1511.02306 Patrick Rall Algorithms from Block Encodings May 2020 6 / 22

Block-encodings Even more general form of circuit: | 0 � | 0 � U | ψ � Patrick Rall Algorithms from Block Encodings May 2020 7 / 22

Block-encodings Even more general form of circuit: | 0 � | 0 � U | ψ � U is a block-encoding of A if: � A � · A = ( � 0 | ⊗ I ) U ( | 0 � ⊗ I ) or U = · · Patrick Rall Algorithms from Block Encodings May 2020 7 / 22

Block-encodings Even more general form of circuit: | 0 � | 0 � U | ψ � U is a block-encoding of A if: � A � · A = ( � 0 | ⊗ I ) U ( | 0 � ⊗ I ) or U = · · Limitation: spectral norm of A at is most 1. Add notion of scale: Patrick Rall Algorithms from Block Encodings May 2020 7 / 22

Block-encodings Even more general form of circuit: | 0 � | 0 � U | ψ � U is a block-encoding of A if: � A � · A = ( � 0 | ⊗ I ) U ( | 0 � ⊗ I ) or U = · · Limitation: spectral norm of A at is most 1. Add notion of scale: U is an α -scaled block-encoding of A if: � A /α � · A /α = ( � 0 | ⊗ I ) U ( | 0 � ⊗ I ) or U = · · Patrick Rall Algorithms from Block Encodings May 2020 7 / 22

New primitives If you have an α -scaled block-encoding of A , you can: | ψ � → A | ψ � Patrick Rall Algorithms from Block Encodings May 2020 8 / 22

New primitives If you have an α -scaled block-encoding of A , you can: | ψ � → A | ψ � | ψ � → A | ψ � � 1 � | A | ψ �| with O | A | ψ �| Patrick Rall Algorithms from Block Encodings May 2020 8 / 22

New primitives If you have an α -scaled block-encoding of A , you can: | ψ � → A | ψ � | ψ � → A | ψ � � 1 � | A | ψ �| with O | A | ψ �| � α � estimate | A | ψ �| with O ε Patrick Rall Algorithms from Block Encodings May 2020 8 / 22