On Quantum and Classical Space-Bounded Processes with Algebraic Transition Amplitudes

• Authors: John Watrous
• Venue: FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
• Year: 1999

We define a class of stochastic processes based on evolutions and measurements of quantum systems, and consider the complexity of predicting their long-term behavior. It is shown that a very general class of decision problems regarding these stochastic processes can be efficiently solved classically in the space-bounded case. The following corollaries are implied by our main result for any space-constructible, not sub-logarithmic space bound s.Any space O(s) uniform family of quantum circuit acting on s qubits and consisting of unitary gates and measurement gates defined in a typical way by matrices of algebraic numbers can be simulated by an unbounded error space O(s) ordinary (i.e., fair-coin flipping) probabilistic Turing machine, and hence by space O(s) uniform classical (deterministic) circuits of depth O(s2) and size 2O(s). The quantum circuits are not required to operate with bounded error and may have depth exponential in s.Any quantum Turing machine running in space s, having arbitrary algebraic transition amplitudes, allowing unrestricted measurements during its computation, and having no restrictions on running time can be simulated by a space O(s) ordinary probabilistic Turing machine in the unbounded error setting.We also obtain the following classical result: Any unbounded error probabilistic Turing machine running in space s that allows algebraic probabilities and algebraic cut-point can be simulated by a space O(s) ordinary probabilistic Turing machine with cut-point 1/2. Our technique for handling algebraic numbers in the above simulations may be of independent interest. It is shown that any real algebraic number can be accurately approximated by a ratio of GapL functions.