The Space Just Above One Clean Qubit

• Main Authors: Jacobs, Dale, Mehraban, Saeed
• Format: Journal Article
• Language: English
• Published: 10-10-2024
• Subjects: Computer Science - Computational Complexity; Physics - Quantum Physics
• DOI: 10.48550/arxiv.2410.08051

Consider the model of computation where we start with two halves of a
$2n$-qubit maximally entangled state. We get to apply a universal quantum
computation on one half, measure both halves at the end, and perform classical
postprocessing. This model, which we call $\frac12$BQP, was defined in STOC
2017 [ABKM17] to capture the power of permutational computations on special
input states. As observed in [ABKM17], this model can be viewed as a natural
generalization of the one-clean-qubit model (DQC1) where we learn the content
of a high entropy input state only after the computation is completed. An
interesting open question is to characterize the power of this model, which
seems to sit nontrivially between DQC1 and BQP. In this paper, we show that
despite its limitations, this model can carry out many well-known quantum
computations that are candidates for exponential speed-up over classical
computations (and possibly DQC1). In particular, $\frac12$BQP can simulate
Instantaneous Quantum Polynomial Time (IQP) and solve the Deutsch-Jozsa
problem, Bernstein-Vazirani problem, Simon's problem, and period finding. As a
consequence, $\frac12$BQP also solves Order Finding and Factoring outside of
the oracle setting. Furthermore, $\frac12$BQP can solve Forrelation and the
corresponding oracle problem given by Raz and Tal [RT22] to separate BQP and
PH. We also study limitations of $\frac12$BQP and show that similarly to DQC1,
$\frac12$BQP cannot distinguish between unitaries which are close in trace
distance, then give an oracle separating $\frac12$BQP and BQP. Due to this
limitation, $\frac12$BQP cannot obtain the quadratic speedup for unstructured
search given by Grover's algorithm [Gro96]. We conjecture that $\frac12$BQP
cannot solve $3$-Forrelation.