Characterizing Quantum Supremacy in Near-Term Devices S. Boixo S. - PowerPoint PPT Presentation

Characterizing Quantum Supremacy in Near-Term Devices S. Boixo S. Isakov, V. Smelyanskiy, R. Babbush, M. Smelyanskiy, N. Ding, Z. Jiang, J.Martinis, H. Neven ICTP August 27th

Quantum Supremacy J. Preskill, 2012 With a quantum device perform a well-defined computational task beyond the capabilities of state-of-the-art classical supercomputers in the near-term without error correction. Not necessarily solving a practical problem.

Approaches to quantum supremacy Optimization of a classical function: Quantum Annealing. Quantum Approximate Optimization Algorithm (E. Farhi et. al.). Non-simulable Hamiltonian Evolution. Variational Quantum Eigensolver (Ground state energy of a Hamiltonian). Approximate sampling from a well defined distribution: Commuting Quantum Circuits (M. Bremner et. al.). BosonSampling. (Aaronson and Arkhipov). Random Universal Circuits. “Randomized benchmarking for complex circuits.”

Requisites for quantum supremacy in the near-term Classically, nothing must work for the computational task, except direct simulation of quantum evolution. Cost exponential in size of Hilbert space. Typical of chaotic systems. Specific figure of merit for the computational task. We should measure the figure of merit up to quantum supremacy frontier. Naturally related to fidelity. Well understood extrapolation of figure of merit beyond the quantum supremacy frontier where it can not be measured. (Unfortunately, we lack witness.) Predictions from theory for figure of merit. Relation to Computational Complexity is a plus. Formal Computational Complexity is asymptotic, requires error correction (Strong Church-Turing Thesis).

Random Universal Quantum Circuits X 1 / 2 / | 0 � H • T • / | 0 � H • • • • / / Y 1 / 2 T | 0 � • • • / / • H T Y 1 / 2 / | 0 � H • T • • / • | 0 � H • T • / / Figure: Vertical lines correspond to controlled-phase gates . Random quantum circuits are examples of quantum chaos. Classically sampling p U ( x ) = | � x | U | 0 � | 2 requires direct simulations. Cost in 2 D exponential in ∝ min ( n , d √ n ) , depth d , qubits n . (With 7 × 7 qubits requires d ≃ 25. ) Good benchmark for quantum computers. New results in computational complexity.

Porter-Thomas distribution (Pseudo-)random circuit U N � | Ψ � = U | 0 � = c i | x i � . j = 1 Sample the output distribution with probabilities p i = | c i | 2 = | � x i | U | Ψ � | 2 . Real and imaginary parts of c i are distributed (quasi) uniformly on a 2 N dimensional sphere (Hilbert space) if the circuit (or Hamiltonian evolution) has sufficient depth (evolution time). The distribution of c i is, up to finite moments, Gaussian with mean 0 and variance ∝ 1 / N (random unitary matrices, delocalization, level repulsion...). Porter-Thomas distribution: Pr ( Np ) = e − Np .

Porter-Thomas distribution Histogram of the output distribution for different values of the two-qubit gate error rate r . 10 1 10 0 10 − 1 Pr( Np ) 10 − 2 r=0 r=0.001 10 − 3 r=0.002 r=0.01 r=0.005 10 − 4 0 1 2 4 6 8 10 Np Figure: Circuit with 5 × 4 qubits (2D lattice) and depth 25.

Verification and uniformity test The PT distribution is very flat: p ( x j ) ∼ 1 / N . The ℓ 1 distance between PT and uniform distribution is � | p ( x j ) − 1 / N | = 2 / e . j If we calculate p ( x j ) given circuit U , we can distinguish these distributions with a constant number of measurements. If we don’t know anything about p ( x j ) (black-box setting) √ we need Θ( N ) measurements. There is no polynomial witness for this sampling problem. This problem is much harder than NP . This is required for near-term (few qubits) supremacy.

Sampling from ideal circuit U Sample S = { x 1 , . . . , x m } of bit-strings x j from circuit U (measurements in the computational basis). � log p U ( x j ) = − m H ( p U ) + O ( m 1 / 2 ) , log Pr U ( S ) = x j ∈ S where H ( p U ) is the entropy of PT � ∞ pN 2 e − Np log p dp = log N − 1 + γ . H ( p U ) = − 0 and γ ≃ 0 . 577.

Sampling with polynomial classical circuit A pcl ( U ) A polynomial classical algorithm A pcl ( U ) produces sample S pcl = { x pcl 1 , . . . , x pcl m } . The probability Pr U ( S pcl ) that this sample S pcl is observed from the output | ψ � of the circuit U is log Pr U ( S pcl ) = − m H ( p pcl , p U ) + O ( m 1 / 2 ) , where N � H ( p pcl , p U ) ≡ − p pcl ( x j | U ) log p U ( x j ) j = 1 is the cross entropy.

Sensitivity to single Pauli error A single Pauli error (almost) destroys the output distribution p U . 8 7 6 No errors 5 Np 4 3 2 One Pauli error (averaged) 1 0 1 N Bit-string index j (same ordering) Figure: Blue line shows sorted probabilities p U ( x j ) (universal quantum chaos distribution, Porter-Thomas). Red line average of a single Pauli error in all different locations, same ordering.

Sampling with polynomial classical circuit A pcl ( U ) (II) We are interested in the average over { U } of random circuits (or chaotic evolutions)   N 1 �  . � � H ( p pcl , p U ) = E U p pcl ( x j | U ) log E U  p U ( x j ) j = 1 Because U is chaotic, Hilbert space has exponential dimension, and A pcl ( U ) is polynomial, we conjecture that p pcl and p U are (almost) uncorrelated (more reasons later). We can take averages independently. � ∞ Ne − Np log p dp = log N + γ . � � − E U log p U ( x j ) ≈ − 0 � � E U H ( p pcl , p U ) = log N + γ ≡ H 0 .

Cross entropy difference The average cross entropy of a polynomial classical algorithm is the same as for a uniform distribution p ( x ) = 1 / N . For algorithm A (quantum or classical of any cost) define the cross entropy difference α ≡ ∆ H ( p A ) ≡ log N + γ − H ( p A , p U ) � 1 � 1 � = N − p A ( x j | U ) log p U ( x j ) . j The cross entropy goes between α = 0 for no correlation, and α = 1 for the ideal circuit.

Cross entropy and fidelity The output of an evolution with fidelity ˜ α is α U | 0 �� 0 | U † + ( 1 − ˜ ρ = ˜ α ) σ U , with p exp ( x ) = � x | ρ | x � = ˜ α p U ( x ) + ( 1 − ˜ α ) � x | σ U | x � . We again conjecture that � x | σ U | x � is uncorrelated with p U ( x ) . α = E U [∆ H ( p exp )] � � � = H 0 + α p U ( x j ) + ( 1 − ˜ ˜ α ) � x j | σ U | x j � x log p U ( x j ) j = H 0 − ˜ α H ( p U ) − ( 1 − ˜ α ) H 0 = ˜ α . The cross entropy α approximates the fidelity ˜ α .

Numerics and theory for realistic 2D circuits Cross entropy difference � and estimated fidelity ◦ . Supremacy frontier r=0 1 . 0 r=0.0005 0 . 8 r=0.001 r=0.002 0 . 6 α 0 . 4 r=0.005 0 . 2 r=0.01 0 . 0 15 20 25 30 35 40 45 50 Number of qubits r is two-qubit gate error rate. α = 1 for chaotic state. d = 25.

Experimental proposal Implement a random universal circuit U (chaotic evolution). 1 Take large sample S exp = { x exp 1 , . . . , x exp m } of bit-strings x in 2 the computational basis ( m ∼ 10 3 − 10 6 ). Compute quantities log p U ( x exp ) with supercomputer. 3 j Cross entropy difference (figure of merit) m α = 1 κ ) + log 2 n + γ ± � log p U ( x exp √ m , κ ≃ 1 , γ = 0 . 577 j m j = 1 Measure and extrapolate α (size, depth, T gates). Fit to theory: α approx. circuit fidelity, chaotic state very sensitive to errors. α ≈ exp ( − r 1 g 1 − r 2 g 2 − r init n − r mes n ) , r 1 , r 2 ≪ 1 one and two-qubit gates Pauli error rates, g 1 , g 2 ≫ 1 number of one and two-qubit gates, r init , r mes ≪ 1 initialization and measurement error rates.

Convergence to chaos Depth required for PT distribution, in 2D is ∝ √ n . 29 . 5 29 . 0 28 . 5 28 . 0 Entropy 25 . 0 24 . 5 27 . 5 24 . 0 27 . 0 23 . 5 23 . 0 26 . 5 22 . 5 0 5 10 15 20 25 30 26 . 0 0 5 10 15 20 25 30 Depth Figure: 2D circuit 7 × 6 qubits. Inset 6 × 6 qubits. Dashed line is known H ( p U ) for PT.

Convergence to chaos (II) Moments of p U converge to PT distribution. 10 19 1 . 10 k=10 10 17 1 . 08 10 15 1 . 06 k=8 � p k � N k − 1 / k ! 10 13 1 . 04 � p k � N k − 1 / k ! 10 11 k=6 1 . 02 k=4 10 9 1 . 00 10 7 k=2 0 . 98 10 5 25 26 27 28 29 30 Depth 10 3 10 1 5 10 15 20 25 30 Depth Figure: Moments � p k � with k = 2 , 4 , 6 , 8 , 10, normalized to 1 for PT distribution. 7 × 6 circuit.

Convergence to chaos (III) 36 34 32 30 Depth 28 26 24 22 20 15 20 25 30 35 40 45 Number of qubits Figure: First cycle such that the entropy remains within 2 − n / 2 of PT entropy.

Complex Ising models from universal circuits As in a path integral, the output amplitude of U is d � s t | U ( t ) | s t − 1 � , � � | s d � = | x � . � x | U | 0 � = { s t } t = 0 where | s t � = ⊗ n j = 1 | s t j � is the computational basis, s t j = ± 1, and U ( t ) are gates at clock cycle t . Gates give Ising couplings between spins s k j , like in path integral QMC. For instance, for X 1 / 2 gates d ( j ) n 1 + s k − 1 s k i π ( x ) = i π 4 H X 1 / 2 j j � � α k . s j 2 2 j = 1 k = 0 j = 1 denotes that a X 1 / 2 gate was applied at where α k qubit j in (clock cycle) k .