Quantum states can be used to encode the information contained in a direction, i.e., in a unit vector. We present the best encoding procedure when the quantum state is made up of N spins (qubits). We find that the quality of this optimal procedure, which we quantify in terms of the fidelity, depends solely on the dimension of the encoding space. We also investigate the use of spatial rotations on a quantum state, which provide a natural and less demanding encoding. In this case we prove that the fidelity is directly related to the largest zeros of the Legendre and Jacobi polynomials. We also discuss our results in terms of the information gain.
Recent work has shown that a simple chain of interacting spins can be used as a medium for high-fidelity quantum communication. We describe a scheme for quantum communication using a spin system that conserves
z spin, but otherwise is arbitrary. The sender and receiver are assumed to directly control several spins each, with the sender encoding the message state onto the larger state space of her control spins. Given that the encoding for the ``zero'' message basis state is chosen to be the all-spin-down state, we show how to find the encoding for the ``one'' basis state that maximizes the fidelity of communication, using a simple method based on the singular-value decomposition. Also, we show that this solution can be used to increase communication fidelity in a rather different circumstance: where no encoding of initial states is used, but where the sender and receiver control exactly two spins each and vary the interactions on those spins over time. The methods presented are computationally efficient, and numerical examples are given for systems having up to 300 spins.
For many important optimization problems we are restricted to approximate solutions in practice due to computational complexity. Distinct from the exact optimization setting, approximate optimization admits performance measures beyond whether the optimum is found, with different tradeoffs and complexity. For MaxCut, a near-unity (ordinary) approximation ratio can coexist with near-zero improvement (gain) over a random cut. For the standard encoding, the unconditional classical MaxCut-Gain hardness gap implies that \emph{any uniformly efficient quantum or hybrid procedure recovering a fixed positive fraction of the optimal classical gain on every input, with at least inverse-polynomial success probability, would place} NP \emph{in} BQP. Such a procedure is therefore believed impossible under standard assumptions. We broadly address where our worst-case barriers do or do not apply across the quantum algorithm landscape. We prove that the barrier survives quantum random access optimization (QRAO) compression and applies between the classical and relaxed optimal values. For every input, a product state attains the classical optimum. Thus the barrier to reaching the classical threshold does not arise from a need for entanglement. For
d∈{2,3} variables per qubit, the known decoder transfers encoded energy gain to decoded mean gain by the exact factor
1/d2. Combining this identity with MaxCut-Gain hardness gives an operational preparation barrier for QRAO. We also construct hard
n-qubit families with relative quantum relaxation excess
Θ(1/n), while the maximally mixed state has energy approximation ratio
1−Θ(1/n), zero encoded energy gain, and hence zero decoded mean gain. Our results separate the effects of relaxation tightness and energy approximation from operational accessibility, motivating more comprehensive accounting in benchmarking and performance assessment.