The Space Just Above One Clean Qubit (2410.08051v1)

Abstract: 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.

• The paper introduces the 1/2BQP model, demonstrating its ability to simulate IQP circuits and solve benchmark problems like Simon’s Problem and Period Finding.
• The paper shows that 1/2BQP offers more computational power than DQC1 while still falling short of full BQP, notably in tasks requiring efficient unitary distinction.
• The paper outlines open problems and future directions, urging further research into extending 1/2BQP’s applications in noisy quantum environments and complex computational tasks.

An Analysis of Computational Power in the $\frac{1}{2}$BQP Model

The paper, "The Space Just Above One Clean Qubit" by Dale Jacobs and Saeed Mehraban, examines an intermediate model of quantum computation termed $\frac{1}{2}$BQP. This model resides between the complexity classes DQC1 and BQP. The authors explore the model's computational power, demonstrating its capabilities and limitations through various quantum algorithms traditionally seen as benchmarks for evaluating quantum advantage over classical counterparts.

Summary of $\frac{1}{2}$BQP

The $\frac{1}{2}$BQP model is introduced as a generalization of the one-clean-qubit model (DQC1). The setup involves starting with half of a maximally entangled state and allowing quantum computation on this half, measuring both parts that culminate in a classical post-processing stage. This framework draws parallels to computation with a state that is initially unknown and is learned only after the computation has been completed.

Comparison with Other Models

Containment within $\frac{1}{2}$BQP

The model can simulate multiple known sub-universal quantum models. For instance, it can effectively simulate circuits in the Instantaneous Quantum Polynomial (IQP) model. Additionally, $\frac{1}{2}$BQP is shown to solve key quantum problems such as Simon’s Problem, Period Finding, and Forrelation—problems known for establishing quantum supremacy over classical computation in BQP vs BPP comparisons.

Limitations

Despite its advantages, $\frac{1}{2}$BQP faces certain limitations. The model cannot efficiently distinguish between unitaries close in trace distance, akin to DQC1. This constraint is crucial when considering applications such as Grover's search algorithm, where $\frac{1}{2}$BQP does not achieve the quadratic speedup characteristic of BQP.

Implications and Open Problems

Theoretical Implications

The theoretical implications include questioning the boundary between what is achievable in $\frac{1}{2}$BQP vs BQP. Although the model performs many quantum tasks, establishing its exact position in the complexity landscape is a nontrivial pursuit. The paper suggests that while $\frac{1}{2}$BQP seems to offer more computational power than DQC1, it falls short of full BQP capability, as evidenced by its inability to extend Forrelation hierarchy to $3$-Forrelation.

Practical Considerations

From a practical standpoint, $\frac{1}{2}$BQP offers potential for quantum computations that are feasible with limited qubit cleanliness, opening avenues for experimentation in noisy quantum environments where subsystems can be reliably checked post-computation.

Future Directions

The authors propose numerous open questions: Can $\frac{1}{2}$BQP implement a full quantum Fourier transform over arbitrary cyclic groups? Does it encompass other intermediate quantum models like BosonSampling, or might it be incomparable to models such as NISQ? Exploring these questions could define more precisely where $\frac{1}{2}$BQP fits into the quantum complexity hierarchy, offering insight into the subtle gradients of quantum computational power.

Conclusion

"The Space Just Above One Clean Qubit" advances our understanding of intermediate quantum computational models. By situating $\frac{1}{2}$BQP between DQC1 and BQP, Jacobs and Mehraban provide a nuanced exploration of the limits and capabilities of quantum complexity classes that leverage entanglement and noisy inputs. This paper enriches the ongoing discourse on the exact nature of computational power within quantum models that are neither classical nor fully quantum, contributing valuable insight into potential pathways for achieving quantum advantage.