We introduce a framework to upper bound the entanglement content of a bipartite quantum state from its spectrum alone. Using linear maps and their inverses, we derive rigorous constraints on the maximal entanglement that can be activated under global unitary transformations. We use as entanglement quantifiers the negativity and the Schmidt number; however, our framework is general and applies to any other entanglement measure. Our approach yields compact analytical sufficient criteria for bounding the entanglement of full-rank states in arbitrary dimensions and reveals new spectral constraints on Schmidt number witnesses.
The work here enables a cost scaling of $\tilde{O}\left( \frac{T ^{1+o(1)} {\rm log}^2 (1/\epsilon)}{\varepsilon}\right)$ and maximum depth $\tilde{O}\left( \frac{T^{1+o(1)} {\rm log} (1/\epsilon)}{\varepsilon}\right)$ for estimating observables after some Hamiltonian evolution $e^{i H T}$ using no extra block-encoding qubits, where $\epsilon$ is the algorithmic error of the target Hamiltonian evolution and $\varepsilon$ is the statistical error from the observable estimation. Throughout, $\tilde{O}$ suppresses factors that are doubly logarithmic in $1/\epsilon$ (i.e. ${\rm polyloglog}(1/\epsilon)$), which are subdominant to the displayed ${\rm polylog}(1/\epsilon)$ factors. This is achieved through product formulas, stable interpolation (Chebyshev), and to calculate the needed fractional queries, cardinal sine interpolation is used. This is an improvement over similar work using interpolation methods which have a superlinear scaling of the form $5^p T^{1+1/p}$, where $p$ is the order of the product formula used and an improvement on required qubits over Quantum Signal Processing methods while maintaining scaling with ${\rm polylog}{1/\epsilon}$. This is accomplished through an optimized and variable (with $T$) order $p$. The slightly better $\log^2(1/\epsilon)$ compared to $\log^3(1/\epsilon)$ comes from the use of cardinal sine interpolation compared to other methods using Chebyshev interpolation and product formulas. The work presented here also provides an alternative to Quantum Signal Processing techniques to calculate observables or amplitudes after a fractional query evolution with a cost $O(T\log{1/\epsilon})$ through this cardinal sine interpolation.
In recent work [1], we proposed a unified notion of nonclassicality that applies to arbitrary processes in quantum theory, including individual quantum states, measurements, and sets thereof. This notion is derived from the principle of generalized noncontextuality, but in a novel manner that applies to individual processes rather than full experiments or theories. In the present work, we develop semidefiniteprogramming-based certificates and witnesses for the nonclassicality of states, sources, measurements, and sets thereof. plement theory-independent approaches based on noncontextuality inequalities. We demonstrate the framework through a variety of explicit examples.
The preparation of quantum states is one of the most fundamental tasks in quantum computing, and a key primitive in many quantum algorithms. Of particular interest to areas such as quantum simulation and linear-system solvers are sparse quantum states, which contain only a small number s of non-zero computational basis states compared to a generic state. In this work, we present an approach that prepares s-sparse states on n qubits, reducing the number of Toffoli gates required compared to prior art. We work in the established framework of first preparing a dense state on a [log(s)]-qubit sub-register, and then mapping this state to the target state via an isometry, with the latter step dominating the cost of the full algorithm. The speed-up is achieved by designing an efficient algorithm for finding and implementing the isometry. The worst-case Toffoli cost of our isometry circuit, which may be viewed as a batched version of an approach by Malvetti et al., is essentially 2s for sufficiently large values of n, yielding roughly a log(s)/2 improvement factor over the state-of-the-art. In numerical benchmarks on randomly chosen states, the cost is closer to s. With the improved isometry circuit, we examine the dense-state preparation step and present ways to optimize the joint cost of both steps, particularly in the case of target states with purely real coefficients, by outsourcing some sub-tasks from the dense-state preparation to the isometry.
The Clifford group plays a central role in quantum information science. It is the building block for many error-correcting schemes and matches the first three moments of the Haar measure over the unitary group-a property that is essential for a broad range of quantum algorithms, with applications in pseudorandomness, learning theory, benchmarking, and entanglement distillation. At the heart of understanding many properties of the Clifford group lies the Clifford commutant: the set of operators that commute with k-fold tensor powers of Clifford unitaries. Previous understanding of this commutant has been limited to relatively small values of k, constrained by the number of qubits n. In this work, we develop a complete theory of the Clifford commutant. Our first result provides an explicit orthogonal basis for the commutant and computes its dimension for arbitrary n and k. We also introduce an alternative and easy-tomanipulate basis formed by isotropic sums of Pauli operators. We show that this basis is generated by products of permutations-which generate the unitary group commutant-and at most three other operators. Additionally, we develop a graphical calculus allowing a diagrammatic manipulation of elements of this basis. These results enable a wealth of applications: among others, we characterize all measurable magic measures and identify optimal strategies for stabilizer property testing, whose success probability also offers an operational interpretation to stabilizer entropies. Finally, we show that these results also generalize to multi-qudit systems with prime local dimension.
