[Paper Review] Quantum Random Access Stored-Program Machines

Qisheng Wang, Mingsheng Ying|arXiv (Cornell University)|Mar 7, 2020

Quantum Computing Algorithms and Architecture28 references4 citations

TL;DR

This paper formally defines Quantum Random Access Machines (QRAMs) and Quantum Random Access Stored-Program Machines (QRASPs) as quantum generalizations of classical RAM and RASP models, establishing their computational equivalence with Quantum Turing Machines (QTMs). It proves that P ⊆ EQRAMP ⊆ EQP ⊆ BQP = BQRAMP, resolving long-standing gaps in the relationships between quantum computation models through a novel standardization technique for QTMs with extended halting schemes.

ABSTRACT

Random access machines (RAMs) and random access stored-program machines (RASPs) are models of computing that are closer to the architecture of real-world computers than Turing machines (TMs). They are also convenient in complexity analysis of algorithms. The relationships between RAMs, RASPs and TMs are well-studied. However, clear relationships between their quantum counterparts are still missing in the literature. We fill in this gap by formally defining the models of quantum random access machines (QRAMs) and quantum random access stored-program machines (QRASPs) and clarifying the relationships between QRAMs, QRASPs and quantum Turing machines (QTMs). In particular, we show that $ extbf{P} \subseteq extbf{EQRAMP} \subseteq extbf{EQP} \subseteq extbf{BQP} = extbf{BQRAMP}$, where $ extbf{EQRAMP}$ and $ extbf{BQRAMP}$ stand for the sets of problems that can be solved by polynomial-time QRAMs with certainty and bounded-error, respectively. At the heart of our proof, is a standardisation of QTM with an extended halting scheme, which is of independent interest.

Motivation & Objective

• To formally define QRAM and QRASP models as quantum extensions of classical RAM and RASP architectures.
• To close the gap in the literature regarding the relationships between quantum computation models, particularly between QRAMs, QRASPs, and QTMs.
• To provide a theoretical foundation for quantum programming by modeling current quantum computers as QRASPs with classical program storage.
• To establish precise complexity class relationships involving EQRAMP, BQRAMP, EQP, and BQP.

Proposed method

• Introduce formal operational and denotational semantics for QRAMs and QRASPs, including address-safe and measurement-postponed variants.
• Develop a standardized QTM model with an extended halting scheme to bridge the gap between QTM and QRAM/QRASP computation models.
• Construct simulations showing that QRAMs and QRASPs simulate each other in polynomial time.
• Use the Solovay-Kitaev algorithm to simulate quantum circuits within QTMs, enabling QTM simulation of QRAMs.
• Prove that a T(n)-time QRAM can be simulated by a QTM in Õ(T(n)^4) time under logarithmic cost and Õ(T(n)^8) under constant cost.
• Establish that a T(n)-time QTM can be simulated by a QRAM in O(T(n)^2 polylog(T(n), 1/ε)) time, under both cost criteria.

Experimental results

Research questions

• RQ1 How can QRAM and QRASP models be formally defined as quantum extensions of classical RAM and RASP architectures?
• RQ2 What is the precise relationship between the computational power of QRAMs, QRASPs, and QTMs?
• RQ3 Can QTMs simulate QRAMs and QRASPs efficiently, and vice versa, under standard cost models?
• RQ4 How do the complexity classes EQRAMP, BQRAMP, EQP, and BQP relate to each other and to classical complexity classes like P?
• RQ5 What techniques are required to reconcile the halting schemes of QTMs with those of QRAMs and QRASPs?

Key findings

• The paper establishes the complexity class hierarchy: P ⊆ EQRAMP ⊆ EQP ⊆ BQP = BQRAMP.
• A T(n)-time QRAM under the logarithmic cost criterion can be simulated by a QTM in Õ(T(n)^4) time.
• A T(n)-time QRAM under the constant cost criterion can be simulated by a QTM in Õ(T(n)^8) time.
• A T(n)-time QTM can be simulated by a QRAM in O(T(n)^2 polylog(T(n), 1/ε)) time, under both cost models.
• QRAMs and QRASPs simulate each other in polynomial time, confirming their computational equivalence.
• A novel standardization technique for QTMs with extended halting schemes is developed, which is of independent theoretical interest.