The Space Just Above One Clean Qubit

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.