Closed Timelike Curves Make Quantum and Classical Computing Equivalent (0808.2669v1)

Abstract: While closed timelike curves (CTCs) are not known to exist, studying their consequences has led to nontrivial insights in general relativity, quantum information, and other areas. In this paper we show that if CTCs existed, then quantum computers would be no more powerful than classical computers: both would have the (extremely large) power of the complexity class PSPACE, consisting of all problems solvable by a conventional computer using a polynomial amount of memory. This solves an open problem proposed by one of us in 2005, and gives an essentially complete understanding of computational complexity in the presence of CTCs. Following the work of Deutsch, we treat a CTC as simply a region of spacetime where a "causal consistency" condition is imposed, meaning that Nature has to produce a (probabilistic or quantum) fixed-point of some evolution operator. Our conclusion is then a consequence of the following theorem: given any quantum circuit (not necessarily unitary), a fixed-point of the circuit can be (implicitly) computed in polynomial space. This theorem might have independent applications in quantum information.

• The paper shows that closed timelike curves enable classical and quantum computers to achieve PSPACE-level computational power.
• It employs Deutsch’s CTC model and fixed-point constraints to simulate looping computations that resolve time paradoxes.
• The findings imply that if CTCs exist, the traditional boundary between classical and quantum complexity would vanish.

Overview of the Computational Power of Closed Timelike Curves

The paper "Closed Timelike Curves Make Quantum and Classical Computing Equivalent" by Scott Aaronson and John Watrous addresses a compelling problem in the intersection of computational complexity and theoretical physics. The authors investigate the computational implications of closed timelike curves (CTCs), hypothetical structures in spacetime that allow for time travel into the past. They demonstrate that CTCs would equalize the computational power of classical and quantum computers, both achieving the capabilities of the complexity class $\mathsf{PSPACE}$.

Computational Complexity and CTCs

The core finding of the paper is that the presence of CTCs would enhance the computational power of both classical and quantum computers to solve problems within $\mathsf{PSPACE}$, a class comprising problems solvable with a polynomial amount of memory. This results in quantum and classical computers being computationally equivalent under the influence of CTCs. The authors leverage Deutsch's model of CTCs, which involves a causal consistency condition that dictates the physical evolution must be a fixed point of a probabilistic or quantum operation.

Achieving $\mathsf{PSPACE}$ Power

The paper highlights several technical routes to demonstrate $\mathsf{PSPACE}$ equivalence. For classical computers with CTCs, the authors use an ability of CTCs to implement a looping strategy over computations, harnessing fixed-point solutions that correspond to $\mathsf{PSPACE}$. This approach involves simulating $\mathsf{PSPACE}$ problems and addressing the Grandfather Paradox through probabilistic resolution strategies.

For quantum computers, borrowing results from linear algebra and complexity theory, the authors illustrate a method to compute fixed points of quantum operators using $\mathsf{PSPACE}$. This technique relies partly on existing fast parallel algorithms for linear algebra. Notably, the authors adopt representations of quantum operations to efficiently compute these fixed points within polynomial space.

Implications and Future Directions

This research implies that if CTCs existed, the delineation between classical and quantum computational power would blur significantly. Practically, this would redefine the landscape of computational problems solvable in polynomial resource bounds, merging them with problems now regarded as space-intensive but time-exponential. Moreover, the results suggest potential new approaches in quantum information, given that understanding fixed-point computations in quantum circuits could motivate algorithms for broader quantum computational problems.

Future explorations might delve into whether partial implementations of CTC-like behavior could practically enhance current computational paradigms. Moreover, the paper opens avenues to investigate computational models with specific constraints, such as bounded CTC registers, and examine their computational capacities.

In conclusion, Aaronson and Watrous' work provides a critical insight into the equivalence of classical and quantum computing power in the hypothetical scenario of CTCs, raising thought-provoking questions about the conceptual boundaries of computation in the presence of exotic physical theories. This research significantly advances our grasp of the implications of integrating complex spacetime structures with computational frameworks.