Dense Quantum Measurement Theory

Quantum measurement is a fundamental cornerstone of experimental quantum computations. The main issues in current quantum measurement strategies are the high number of measurement rounds to determine a global optimal measurement output and the low success probability of finding a global optimal measurement output. Each measurement round requires preparing the quantum system and applying quantum operations and measurements with high-precision control in the physical layer. These issues result in extremely high-cost measurements with a low probability of success at the end of the measurement rounds. Here, we define a novel measurement for quantum computations called dense quantum measurement. The dense measurement strategy aims at fixing the main drawbacks of standard quantum measurements by achieving a significant reduction in the number of necessary measurement rounds and by radically improving the success probabilities of finding global optimal outputs. We provide application scenarios for quantum circuits with arbitrary unitary sequences, and prove that dense measurement theory provides an experimentally implementable solution for gate-model quantum computer architectures.

A.

A.2 Notations
The notations of the manuscript are summarized in Table A.1.
Pr (0) = Pr (1) = 0.5, associated with the measurement of the i-th quantum state of the output quantum system, while M B is a quantum measurement in the computational basis z * Global optimal measurement output.
L Number of unitary gates in the QG quantum circuit.
where P is a generalized Pauli operator formulated by a tensor product of Pauli operators {σ X , σ Y , σ Z }, while θ i is referred to as the gate parameter associated to U i (θ i ).

C
Classical objective function of a computational problem fed into the quantum computer.
P Generalized Pauli operator formulated by the tensor product of Pauli operators {σ X , σ Y , σ Z }.

|X
An n-length input quantum system.

X
Classical representation of |X .

|Y
An n-length output quantum system, An actual setting of the unitaries of QG at a particular computational basis B, to provide output |G = U ( θ )|S , such that U ( θ )|S = U ( θ)|X .
θ L-dimensional vector of the gate parameters of U ( θ ).

B
Computational basis, selected such that L 0 (S) = K, K n, holds for the L 0 -norm of S, where S is a classical representation of |S . Pr R 0 (z * ) Probability of finding the global optimal output z * via R 0 standard measurement rounds.
Pr R (z * ) Probability of finding the global optimal z * via R dense measurement rounds.
Pr R 0 C (z * ) Probability of finding the global optimal C (z * ) via R 0 standard measurement rounds.
Pr R C (z * ) Probability of finding the global optimal C (z * ) via R dense measurement rounds. β C An n-length vector, β C = (b 1 , . . . , b n ) T .

S
Recovered computational basis vector S from Y = β C S via P .
is the measurement result vector of the m-th round.
ξ Error probability of finding z * at the end of the R rounds, Pr (z = z * ) = ξ.

Υ A subset.
H Hermitian matrix.
V, W, D Parameters of the dense measurement procedure.
ϕ kl Inner product of two normalized columns v k and v l , as is a subset of R elements selected uniform at random from all subsets of [n] of cardinality R, |Q R | = R. ξ * Error probability associated with the selection of rows uniformly and independently at random from U ( θ).