[Paper Review] Counting, Fanout, and the Complexity of Quantum ACC

Frederic Green, Steven Homer+2 more|arXiv (Cornell University)|Jun 4, 2001

Quantum Computing Algorithms and Architecture17 references4 citations

TL;DR

This paper introduces quantum analogs of classical circuit classes, defining ${\sf{QAC}}^{0}$ and ${\sf{QACC}}[q]$ for quantum circuits with constant depth and unbounded fan-in. It proves that fanout gates enable construction of ${\rm MOD}_q$ gates in constant depth, showing ${\sf{QAC}}_{\rm wf}^{0} = {\sf{QACC}}[q] = {\sf{QACC}}$ for all $q > 1$, demonstrating a strict quantum advantage over classical ${\sf{ACC}}[q]$ classes.

ABSTRACT

We propose definitions of $\QAC^0$, the quantum analog of the classical class $\AC^0$ of constant-depth circuits with AND and OR gates of arbitrary fan-in, and $\QACC[q]$, the analog of the class $\ACC[q]$ where $\Mod_q$ gates are also allowed. We prove that parity or fanout allows us to construct quantum $\MOD_q$ gates in constant depth for any $q$, so $\QACC[2] = \QACC$. More generally, we show that for any $q,p > 1$, $\MOD_q$ is equivalent to $\MOD_p$ (up to constant depth). This implies that $\QAC^0$ with unbounded fanout gates, denoted $\QACwf^0$, is the same as $\QACC[q]$ and $\QACC$ for all $q$. Since $\ACC[p] e \ACC[q]$ whenever $p$ and $q$ are distinct primes, $\QACC[q]$ is strictly more powerful than its classical counterpart, as is $\QAC^0$ when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for $\QACC^0$ in terms of related language classes. We define classes of languages $\EQACC$, $\NQACC$ and $\BQACC_{ ats}$. We define a notion of $\log$-planar $\QACC$ operators and show the appropriately restricted versions of $\EQACC$ and $\NQACC$ are contained in $¶/\poly$. We also define a notion of $\log$-gate restricted $\QACC$ operators and show the appropriately restricted versions of $\EQACC$ and $\NQACC$ are contained in $\TC^0$.

Motivation & Objective

• To define and formalize quantum analogs of classical circuit classes ${\sf{AC}}^0$ and ${\sf{ACC}}[q]$, specifically ${\sf{QAC}}^{0}$ and ${\sf{QACC}}[q]$.
• To investigate whether quantum circuits with fanout or ${\rm MOD}_q$ gates can achieve greater computational power than their classical counterparts.
• To establish upper bounds on quantum language classes ${\sf{EQACC}}$, ${\sf{NQACC}}$, and ${\sf{BQACC}}_{{\bf Q}}$ by restricting circuit structure.
• To explore the relationship between quantum circuit depth, fanout, and the ability to compute modular counting functions.

Proposed method

• Define ${\sf{QAC}}^{0}$ as constant-depth quantum circuits with one-qubit gates, Toffoli gates, and no fanout; define ${\sf{QAC}}_{\rm wf}^{0}$ with fanout gates.
• Define ${\sf{QACC}}[q]$ as ${\sf{QAC}}^{0}$ extended with quantum ${\rm MOD}_q$ gates, and ${\sf{QACC}} = \bigcup_q {\sf{QACC}}[q]$.
• Prove that the ability to create a cat state (via fanout) enables constant-depth construction of $n$-ary parity gates.
• Show that for any $q > 1$, ${\rm MOD}_q$ gates are equivalent to ${\rm MOD}_p$ gates in constant depth, implying ${\sf{QACC}}[q] = {\sf{QACC}}[p]$.
• Introduce log-planar and log-gate-restricted ${\sf{QACC}}$ operators to bound the complexity of language classes.
• Use tensor graph representations with color-consistent paths to simulate amplitudes and prove containment in ${\sf{P}}/{\sf{poly}}$ and ${\sf{TC}}^0$.

Experimental results

Research questions

• RQ1 Can fanout gates be constructed in constant depth within ${\sf{QAC}}^{0}$, or is ${\sf{QAC}}^{0} \subset {\sf{QAC}}_{\rm wf}^{0}$?
• RQ2 Is ${\sf{QAC}}_{\rm wf}^{0}$ equal to ${\sf{QTC}}^{0}$, i.e., can quantum threshold gates be built in constant depth?
• RQ3 Are the quantum language classes ${\sf{EQACC}}$, ${\sf{NQACC}}$, and ${\sf{BQACC}}_{{\bf Q}}$ contained in ${\sf{TC}}^0$ or ${\sf{P}}/{\sf{poly}}$?
• RQ4 Can classical proof techniques for ${\sf{ACC}}$ (e.g., Valiant-Vazirani, Toda polynomials) be adapted to the quantum setting?
• RQ5 What is the exact complexity of fixed-depth quantum circuits, such as depth-2 ${\sf{QACC}}$ circuits?

Key findings

• Fanout gates in constant depth allow the construction of $n$-ary parity gates, which implies that ${\sf{QAC}}_{\rm wf}^{0}$ can compute any ${\rm MOD}_q$ function in constant depth.
• For any $q > 1$, ${\sf{QACC}}[q] = {\sf{QACC}}$, meaning the choice of $q$ does not affect the computational power of the class.
• The class ${\sf{QAC}}_{\rm wf}^{0}$ is equal to ${\sf{QACC}}[q]$ for all $q > 1$, so ${\sf{QAC}}_{\rm wf}^{0} = {\sf{QACC}}$.
• The language classes ${\sf{EQACC}}_{\rm pl}^{\log}$ and ${\sf{NQACC}}_{\rm pl}^{\log}$ are contained in ${\sf{P}}/{\sf{poly}}$ under log-planar restrictions.
• The language classes ${\sf{EQACC}}_{\rm gates}^{\log}$ and ${\sf{NQACC}}_{\rm gates}^{\log}$ are contained in ${\sf{TC}}^0$ under log-gate-restricted models.
• Under polynomial-time uniformity assumptions, $p$-uniform ${Σ}^{\rm NQACC}_{\rm pl}^{\log}$ is in ${\sf{P}}$, and $p$-uniform ${Σ}^{\rm NQACC}_{\rm gates}^{\log}$ is in $p$-uniform ${\sf{TC}}^0$.