We study the discrimination of Boolean memory configurations via a fixed Universal QRAM (U-QRAM) interface. Given query access to a quantum memory storing an unknown Boolean function $f:[N]\to\{0,1\}$, we ask: what can be inferred about the bias class of $f$ (its imbalance from $1/2$, up to complement symmetry) using coherent, addressable queries? We show that for exact-weight bias classes, the induced single-query ensemble state on the address register has a two-eigenspace structure that yields closed-form expressions for the single-copy Helstrom-optimal measurement and success probability. Because complementing $f$ changes the state $|ψ\rangle$ only by a global phase, hypotheses $p$ and $1-p$ are information-theoretically identical in this model; thus the natural discriminand is the phase-bias magnitude $|μ|$ (equivalently $μ^2$). This goes beyond the perfect-discrimination case of Deutsch-Jozsa and complements exact-identification settings such as Bernstein-Vazirani.
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9 pages, 0 figures, includes supplementary Qiskit implementation
We study the discrimination of Boolean memory configurations via a fixed Universal QRAM (U-QRAM) interface. Given query access to a quantum memory storing an unknown Boolean function f:[N]→{0,1}, we ask: what can be inferred about the bias class of f (its imbalance from 1/2, up to complement symmetry) using coherent, addressable queries? We show that for exact-weight bias classes, the induced single-query ensemble state on the address register has a two-eigenspace structure that yields closed-form expressions for the single-copy Helstrom-optimal measurement and success probability. Because complementing f changes the state ∣ψ⟩ only by a global phase, hypotheses p and 1−p are information-theoretically identical in this model; thus the natural discriminand is the phase-bias magnitude ∣μ∣ (equivalently μ2). This goes beyond the perfect-discrimination case of Deutsch-Jozsa and complements exact-identification settings such as Bernstein-Vazirani.
Universal QRAM (U-QRAM) is a fixed, data-independent unitary interface that implements coherent random-access reads relative to a designated computational "truth-table" basis on the memory register. This work studies restricted-access inference: the memory register is persistent but inaccessible, while an experimenter may prepare and measure only accessible registers and may invoke the fixed read interaction.
Allowing the memory to be in an arbitrary quantum state (pure or mixed, possibly entangled with an inaccessible reference system, or a coherent superposition of truth tables), we establish a sharp, protocol-independent limitation of read-only access. For any finite-query protocol -- including arbitrary accessible ancillas, intermediate measurements, adaptivity, and general CPTP processing between queries -- the induced output state on the accessible registers depends on the memory state only through its diagonal in the truth-table basis. Equivalently, read-only access factors through dephasing (pinching) in that basis; coherences between distinct truth tables are operationally invisible.
Consequently, every memory-hypothesis testing task reduces to a standard state-discrimination problem on the accessible registers, and the minimum-error optimal measurement is characterized by Helstrom theory. We illustrate the framework with three explicit examples: (i) the phase-kickback reduction recovering the one-query Bernstein-Vazirani/Deutsch-Jozsa geometry, (ii) a minimal Helstrom instance with optimal success probability 3/4, and (iii) perfect indistinguishability of relative phases in entangled truth-table superpositions.
Quantum-state filtering is a variant of the unambiguous state discrimination problem: the states are grouped in sets and we want to determine to which particular set a given input state belongs. The simplest case, when the N given states are divided into two subsets and the first set consists of one state only while the second consists of all of the remaining states, is termed quantum-state filtering. We derived previously the optimal strategy for the case of N nonorthogonal states, ${\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$, for distinguishing $\ensuremath{\mid}{\ensuremath{\psi}}_{1}⟩$ from the set ${\ensuremath{\mid}{\ensuremath{\psi}}_{2}⟩,\dots{},\ensuremath{\mid}{\ensuremath{\psi}}_{N}⟩}$ and the corresponding optimal success and failure probabilities. In a previous paper [Phys. Rev. Lett. 90, 257901 (2003)], we sketched an appplication of the results to probabilistic quantum algorithms. Here we fill the gaps and give the complete derivation of the probabilstic quantum algorithm that can optimally distinguish between two classes of Boolean functions, that of the balanced functions and that of the biased functions. The algorithm is probabilistic, it fails sometimes but when it does it lets us know that it did. Our approach can be considered as a generalization of the Deutsch-Jozsa algorithm that was developed for the discrimination of balanced and constant Boolean functions.
Quantum random access memory (QRAM) is a central primitive for coherent data access in quantum algorithms, yet it remains controversial in practice because the wall-clock cost of "one lookup" can hide routing depth, control overhead, and geometric constraints. We present a universal QRAM construction (U-QRAM) in which the database is a physical memory register that participates in the lookup unitary as quantum control. This yields a single fixed, data-independent lookup unitary on A⊗D⊗M that is correct for all basis-encoded databases.
Our first contribution is an explicit, exact Hamiltonian realization: U-QRAM equals a single time-independent evolution UQRAM=exp(−iHtot) where Htot is a sum of mutually commuting projector terms, one per memory cell, and the construction avoids control-dependent phase ambiguities. Our second contribution is an architectural sharpening: under unary (one-hot, mode-addressed) encoding of the address, each cell term becomes uniform and 3-local, delineating the most direct path toward constant-latency interpretations. We keep claims conservative by separating latency from work and stating explicit hardware assumptions required for constant wall-clock queries.
Quantum random access memory (QRAM) is a central primitive for coherent data access in quantum algorithms, yet it remains controversial in practice because the wall-clock cost of "one lookup" can hide routing depth, control overhead, and geometric constraints. We present a universal QRAM construction (U-QRAM) in which the database is a physical memory register that participates in the lookup unitary as quantum control. This yields a single fixed, data-independent lookup unitary on A ⊗ D ⊗ M that is correct for all basis-encoded databases. Our first contribution is an explicit, exact Hamiltonian realization: U-QRAM equals a single time-independent evolution U_QRAM = exp(-iH_tot) where H_tot is a sum of mutually commuting projector terms, one per memory cell, and the construction avoids control-dependent phase ambiguities. Our second contribution is an architectural sharpening: under unary (one-hot, mode-addressed) encoding of the address, each cell term becomes uniform and 3-local, delineating the most direct path toward constant-latency interpretations. We keep claims conservative by separating latency from work and stating explicit hardware assumptions required for constant wall-clock queries.
We introduce a new state discrimination problem in which we are given additional information about the state after the measurement, or more generally, after a quantum memory bound applies. The following special case plays an important role in quantum cryptographic protocols in the bounded storage model: Given a string x encoded in an unknown basis chosen from a set of mutually unbiased bases (MUBs), you may perform any measurement, but then store at mostqqubits of quantum information, and an unlimited amount of classical information. Later on, you learn which basis was used. How well can you compute a function f(x) of x, given the initial measurement outcome, the q qubits, and the additional basis information? We first show a lower bound on the success probability for any balanced function, and any number of mutually unbiased bases, beating the naive strategy of simply guessing the basis. We then show that for two bases, any Boolean function f(x) can be computed perfectly if you are allowed to store just a single qubit, independent of the number of possible input strings x. However, we show how to construct three bases, such that you need to store all qubits in order to compute f(x) perfectly. We then investigate how much advantage the additional basis information can give for a Boolean function. To this end, we prove optimal bounds for the success probability for the AND and the XOR function for up to three mutually unbiased bases. Our result shows that the gap in success probability can be maximal: without the basis information, you can never do better than guessing the basis, but with this information, you can compute f(x) perfectly. We also give an example where the extra information does not give any advantage at all.
We introduce a new state discrimination problem in which we are given additional information about the state after the measurement, or more generally, after a quantum memory bound applies. In particular, the following special case plays an important role in quantum cryptographic protocols in the bounded storage model: Given a string x encoded in an unknown basis chosen from a set of mutually unbiased bases, you may perform any measurement, but then store at most q qubits of quantum information. Later on, you learn which basis was used. How well can you compute a function f(x) of x, given the initial measurement outcome, the q qubits and the additional basis information? We first show a lower bound on the success probability for any balanced function, and any number of mutually unbiased bases, beating the naive strategy of simply guessing the basis. We then show that for two bases, any Boolean function f(x) can be computed perfectly if you are allowed to store just a single qubit, independent of the number of possible input strings x. We then investigate how much advantage the additional basis information can give for a Boolean function. To this end, we prove optimal bounds for the success probability for the AND and the XOR function for up to three mutually unbiased bases. Our result shows that the gap in success probability can be maximal: without the basis information, you can never do better than guessing the basis, but with this information, you can compute f(x) perfectly. We also exhibit an example where the extra information does not give any advantage at all. Interestingly, choosing the string x uniformly at random is not always the hardest distribution, even when f is balanced.