Notes about some of the contributions to the IEEE’s Quantum Week 2021. - December, 2021

Quantum Approximate Optimization

About Pranav Gokhale’s talk at QCE21 [M16]:

  • initial optimism:

    • A Quantum Approximate Optimization Algorithm [35]: QAOA applied to a bound occurence constraint problem, [v1] beats the best known classical algorithm, but [v2] classical algo. have been improved ever since

    • Quantum Supremacy through the QAOA [38]

  • but more recently:

    • QAOA for Max-Cut requires hundreds of qubits for quantum speed-up [52] -> classical \(\textrm{akmaxsat}\) in seconds while QAOA in days (for sparse graphs)

    • Classical and Quantum Bounded Depth Approximate Algorithm [57] -> local classical MAX-3-LIN-2 scales better then QAOA

    • Bounds on approximating Max kXOR with quantum and classical local algorithms [89] -> QAOA beats classical algorithms, but very far away from “Parisi limit” theoretical benchmark

  • noise issue:

    • Noise-Induced Barren Plateaus in VQAs [130], -> increasing the circuit depth makes gradients vanish for points more far away from solution

    • Quantifying the impact of precision errors on QAOA [109]

  • optimism on dense (hyper)graphs:

    • \(\textrm{akmaxsat}\)’s runtime increases exponentially with graph density

    • Optimized fermionic SWAP networks […] for QAOA [56]

  • more optimism:

    • The QAOA and the Sherrington-Kirkpatrick Model at Infinite Size [37] -> from depth p = 11 on, advantage (?)

    • Obstacles to State Preparation and Variational Optimization from Symmetry Protection, IBM/TUM (Alexander Kliesch, Robert Koenig) [22] -> run QAOA recursively on subgraphs

Quantum Kernel Machines

More to digest: with a focus on quantum kernel machines [M21] [M30] [M35]:

  • Quantum embeddings for machine learning, Lloyd & Schuld (2020) [79]

    • about the Hilbert space of the quantum system being a natural space for kernel machines

  • Machine learning of high dimensional data on a noisy quantum processor, FermiLab/Google (2021) [104]

    • use classical data to compute a quantum kernel matrix, then feed this to a classical SVM

    • beyond classical advantage to be found in an “expressive kernel that is classicaly hard to compute”, rather than in speed up (may be one day with quantum error correction)

    • barren plateau problems i.e. regions with vanishing gradients

    • Google Rainbow chip with 23 qubits

    • “fixed shot budget” (i.e. optimization is essential)

  • Kernel Matrix Completion for Offline Quantum-Enhanced Machine Learning, IBM (2021) [96]

    • streaming data: a challenge for quantum kernels

    • matrix completion by a graph-theory-based algorithms using Positive Semidefinite Matrix Completion [126]

    • once the overlap exceeds the rank of the extended matrix, perfect completion is possible: about guessing the rank a priori


Neutral Atoms

About [M38].

  • neutral atoms trapped in an array of optical tweezers (square or triangular lattice, arbitrary patterns)

  • atoms encoded in TLS

  • laser is tuned to drive a coherent transition between the two energy levels

    • incl. detuning (wrt. Rabi frequency)

  • Rydberg states, highly excited electronic states - two atoms will interact through large dipole interaction

  • Rydberg blockade: states get coupled to the entangled Bell state \(\ket{\phi_+}\)

    • Many-body physics with individually controlled Rydberg atoms [R6]

  • measurement through push-out beams and then fluorescence image of the remaining atoms

  • analog vs. digital

    • analog: shine all atoms with the same laser, continuously control the Hamiltonian, of all the qubits at the same time ; and measured at the end

      • a tunable Ising Hamiltonian

    • digital: usual circuit, local operations on specific qubits

      • qubits encoded in 2 hyperfine ground states

      • atoms don’t interact when not in a Rydberg state: no interaction term in the Hamiltonian

      • with one resonant pulse (combined with a change of phase of the laser), any arbitrary single-qubit gate can be performed

      • multi-qubits gates: atoms a brought briefly to the Rydberg state to exploit the Rydberg blockade

      • controlled Z gate (CZ), see below

    • no equivalence of the analog approach as a circuit


About [M31].

  • Continuous-variable quantum information, and the measurement-based computing model [L2].

    • A One-Way Quantum Computer [110]

    • Universal Quantum Computation with Continuous-Variable Cluster States [92]

    • starting from a pre-entangled cluster state, all qubits in \(\ket{0} + \ket{1}\), measurements on individual qubits act as qubit operations

  • Continuous-variable quantum computing in the quantum optical frequency comb [106]