Randomized measurements

• “A quantum state is repeatedly prepared and measured in a randomly chosen basis.

  • Then a classical computer processes the measurement outcomes to estimate the desired property.

• The randomization of the measurement procedure has distinct advantages;

  • for example, a single data set can be employed multiple times to pursue a variety of applications,

  • and imperfections in the measurements are mapped to a simplified noise model that can more easily be mitigated.

  • Use cases that have already been realized in quantum devices: Hamiltonian simulation tasks, probes of quantum chaos, measurements of nonlocal order parameters, and comparison of quantum states produced in distantly separated laboratories.”

• “With sufficiently many repetitions, it is possible to completely characterize an n-qubit state by means of full Quantum state tomography;

  • but this task is hopelessly inefficient, requiring a number of experiments exponential in n, and an amount of classical postprocessing of the experimental results which is also exponentially large.

• A far less complete description of the state is adequate for many purposes. The number of experiments and the amount of classical processing needed can be drastically reduced.”

• “Rather than tailoring the measurements we perform in the lab to the particular properties we wish to study, we can instead repeatedly perform measurements which are randomly sampled from a fixed ensemble.

  • And then adapt the classical postprocessing of the measurement outcomes to the particular task at hand.

  • This randomized measurement strategy can be surprisingly powerful. Even when the measurements are simple enough to be performed with adequate precision using today’s noisy quantum platforms.

  • A particularly simple procedure is to measure each qubit in a randomly chosen basis. By repeating this procedure of order log(L) times, and using only efficient classical postprocessing, we can accurately estimate the expectation values of any L local operators – the number of experiments needed does not depend on the total number of qubits.”

• “A randomized measurement (RM) on N qubits consists in the following steps:

  • (i) Prepare an n-system state ρ.

  • (ii) Apply on each system a local unitary selected at random.

    • The individual single-qubit rotations Un (n=1,…,N) are sampled from ensembles of single-qubit unitary operations which evenly cover the Bloch sphere of each qubit.

    • Examples of such unitary designs include the single-qubit Clifford group, as well as the full unitary group U(2) encompassing all single-qubit transformations.

    • For the sake of simplicity, we also assume that the single-qubit rotations Ui are sampled from the discrete ensemble that randomly permutes Pauli matrices {X,Y,Z} (the Clifford group). That is, UnZUn† = Wn ∈ {X,Y,Z} for 1 ≤ n ≤ N.”

  • (iii) Lastly, a projective measurement in the computational basis {

    s⟩} is performed.

    • outcome bit string s = (s1,…,sN) and sn ∈ {0,1} for n = 1,…,N.

  • Steps (i)-(iii) are then repeated K times with a fixed unitary U.

  • Subsequently, the entire procedure is repeated with M independently sampled unitaries U such that in total M · K experimental runs are performed.”

• “We repeat K > 1 times.

  • This is equivalent to measuring a random string of Pauli observables, U1\dag Z U1” \ox … \ox Un\dag Z U = W1 \ox … \ox WN, a total of K times.

  • This allows to approximate the expectation value tr(W1\ox … \ox WN \rho).

  • As well as compatible subsystem marginals, e.g. tr(W1 \ox … Wl …). Other Pauli expectation values are off limits, though.”

• Applications:

  • Estimate the overlap of two quantum states produced in separate laboratories far apart from one another.

  • Probe chaotic quantum dynamics by measuring out-of-time-order correlation functions.

  • Quantify quantum entanglement by measuring entropy and other entanglement measures.

  • Estimate the expectation value and variance of a local Hamiltonian.”

  • Quantum metrology;

  • “Remarkably, randomized measurements give access not only to observables, but also polynomial functionals of the density matrix. In fact, RMs were first envisioned to estimate the purity P2 = tr(ρ^2)”

• “AI with quantum-to-classical converters:

  • we envision that classical machines may one day achieve a powerful ability to predict the behavior of the quantum world as well.

  • Predict properties of exotic quantum systems that have not previously been realized in the physical lab

    • “Beyond qubits and qudits, programmable quantum many-body systems can be engineered with bosonic or fermionic particles as basic constituents.

      • Ultracold fermionic atoms in optical lattices described by a 2D Fermi Hubbard model with repulsive interactions.

      • Single-site control and site-resolved single-shot readout via a quantum gas microscope.”

  • Design better quantum computers

  • Discover new physical phenomena.

References

• Preskill2022;