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The accurate estimation of quantum observables is a fundamental task in science. Here, we introduce a measurement scheme that adaptively modifies the estimator based on previously obtained data. Our algorithm, which we call AEQuO, allows for overlap in the subsets of Pauli operators that are measured simultaneously, thereby maximizing the amount of information gathered in the measurements. The adaptive estimator comes in two variants: a greedy bucket-filling algorithm with good performance for small problem instances, and a machine learning-based algorithm with more favorable scaling for larger instances. The measurement configuration determined by these subroutines is further post-processed in order to lower the error on the estimator. We test our protocol on chemistry Hamiltonians as well as many-body Hamiltonians with different interaction ranges. In all cases, our algorithm provides error estimates that improve on state-of-the-art methods based on various grouping techniques or randomized measurements.

We investigate the problem of synthesizing T-depth optimal quantum circuits over the Clif-ford+T gate set. The latter is a universal fault-tolerant gate set where the cost of fault-tolerantly implementing the non-Clifford T gate is the most expensive using state-of-the-art techniques such as magic state distillation in surface codes.First, we develop a provable algorithm to synthesize depth optimal circuits using nested meetin-the-middle (MITM) techniques, as have been used by Mosca and Mukhopadhyay [MM20] to synthesize T-count optimal circuits -this algorithm has time and space complexity

An important part of reaping computational advantage from a quantum computer is to reduce the quantum resources needed to implement a desired quantum algorithm. Quantum algorithms that are too large to be practical on noisy intermediate scale quantum devices will require fault-tolerant error correction. This work focuses on reducing the physical cost of implementing quantum algorithms when using the state-of-the-art fault-tolerant quantum error correcting codes, in particular, those for which implementing the T gate consumes vastly more resources than the other gates in the gate set. More specifically, in this paper we consider the group of unitaries that can be exactly implemented by a quantum circuit consisting of the Clifford + T gate set. The Clifford + T gate set is a universal gate set and in this group, using state-of-the-art surface codes, the T gate is by far the most expensive component to implement fault-tolerantly. So it is important to minimize the number of T gates necessary for a fault-tolerant implementation. Our primary interest is to compute a circuit for a given n-qubit unitary U, using the minimum possible number of T gates (called the T-count of unitary U). We consider the problem COUNT-T, the optimization version of which aims to find the T-count of U. In its decision version the goal is to decide if the T-count is at most some positive integer m. Given an oracle for COUNT-T, we can compute a T-count-optimal circuit in time polynomial in the T-count and dimension of U. We give a provable classical algorithm that solves COUNT-T (decision) in time O N 2(c−1) m c poly(m, N) and space O N 2 m c poly(m, N) , where N = 2 n and c 2. This gives a space-time trade-off for solving this problem with variants of meet-in-the-middle techniques. We also introduce an asymptotically faster multiplication method that shaves a factor of N 0.7457 off of the overall complexity. Lastly, beyond our improvements to the rigorous algorithm, we give a heuristic algorithm that outputs a T-count-optimal circuit and has space and time complexity poly(m, N), under some assumptions. In our heuristic algorithm we developed a novel way of pruning the search space. While our heuristic method still scales exponentially with the number of qubits (though with a lower exponent), there is a large improvement by going from exponential to polynomial scaling with m. We implemented our heuristic algorithm with up to 4 qubit unitaries and obtained a significant improvement in time. For all benchmark and random unitaries we studied, the T-count returned by our algorithm is at most the T-count of their circuits shown in previous papers. iπ 4 P.Roughly speaking, each R(P) can be implemented with a circuit consisting of only one T gate. More detail has been given in section 2.2. Operator P is an n-qubit non-identity Pauli operator (defined in section A.1). We developed a fast algorithm in section 3 that computes W in time O(N 4 ). Currently the fastest algorithm

We investigate the problem of synthesizing T-depth optimal quantum circuits for exactly implementable unitaries over the Clifford+T gate set. We construct a subset, $${{\mathbb{V}}}_{n}$$ V n , of T-depth 1 unitaries. T-depth-optimal decomposition of unitary U is $${e}^{i\phi }\left({\prod }_{i}{V}_{i}\right)C$$ e i ϕ ∏ i V i C , $${V}_{i}\in {{\mathbb{V}}}_{n}$$ V i ∈ V n , C is Clifford and $$| {{\mathbb{V}}}_{n}| \,\le \,n\cdot {2}^{5.6n}$$ ∣ V n ∣ ≤ n ⋅ 2 5.6 n . We use nested meet-in-the-middle technique to synthesize provably depth-optimal and T-depth-optimal circuits. For the latter, we achieve space and time complexity $$O({({4}^{{n}^{2}})}^{\lceil d/c\rceil })$$ O ( ( 4 n 2 ) ⌈ d / c ⌉ ) and $$O({({4}^{{n}^{2}})}^{(c-1)\lceil d/c\rceil })$$ O ( ( 4 n 2 ) ( c − 1 ) ⌈ d / c ⌉ ) respectively (d is the minimum T-depth, c ≥ 2 a constant). The previous best algorithm had complexity $$O({({3}^{n}\cdot {2}^{k{n}^{2}})}^{\lceil \frac{d}{2}\rceil }\cdot {2}^{k{n}^{2}})$$ O ( ( 3 n ⋅ 2 k n 2 ) ⌈ d 2 ⌉ ⋅ 2 k n 2 ) (k > 2.5 a constant). We design a more efficient algorithm with space and time complexity poly(n, 25.6n, d) (or $${{{\rm{poly}}}}({n}^{\log n},{2}^{5.6n},d)$$ poly ( n log n , 2 5.6 n , d ) with weaker assumptions). The claimed efficiency, optimality depends on conjectures.