Range Avoidance (AVOID) is a total search problem where, given a Boolean circuit $C\colon\{0,1\}^n\to\{0,1\}^m$, $m>n$, the task is to find a $y\in\{0,1\}^m$ outside the range of $C$. For an integer $k\geq 2$, $\mathrm{NC}^0_k$-AVOID is a special case of AVOID where each output bit of $C$ depends on at most $k$ input bits. While there is a very natural randomized algorithm for AVOID, a deterministic algorithm for the problem would have many interesting consequences. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high formula complexity, rigid matrices, and optimal linear codes, reduce to $\mathrm{NC}^0_4$-AVOID, thus establishing conditional hardness of the $\mathrm{NC}^0_4$-AVOID problem. On the other hand, $\mathrm{NC}^0_2$-AVOID admits polynomial-time algorithms, leaving the question about the complexity of $\mathrm{NC}^0_3$-AVOID open.
We give the first reduction of an explicit construction question to $\mathrm{NC}^0_3$-AVOID. Specifically, we prove that a polynomial-time algorithm (with an $\mathrm{NP}$ oracle) for $\mathrm{NC}^0_3$-AVOID for the case of $m=n+n^{2/3}$ would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits.
We also give deterministic polynomial-time algorithms for all $\mathrm{NC}^0_k$-AVOID problems for $m\geq n^{k-1}/\log(n)$. Prior work required an $\mathrm{NP}$ oracle, and required larger stretch, $m \geq n^{k-1}$.
In this paper, we study the computational complexity of the commutative determinant polynomial computed by a class of set-multilinear circuits which we call regular set-multilinear circuits. Regular set-multilinear circuits are commutative circuits with a restriction on the order in which they can compute polynomials. A regular circuit can be seen as the commutative analogue of the ordered circuit defined by Hrubes,Wigderson and Yehudayoff [HWY10]. We show that if the commutative determinant polynomial has small representation in the sum of constantly many regular set-multilinear circuits, the…
We consider the hardness of computing additive approximations to output probabilities of random quantum circuits. We consider three random circuit families, namely, Haar random, $p=1$ QAOA, and random IQP circuits. Our results are as follows. For Haar random circuits with $m$ gates, we improve on prior results by showing $\mathsf{coC_=P}$ hardness of average-case additive approximations to an imprecision of $2^{-O(m)}$. Efficient classical simulation of such problems would imply the collapse of the polynomial hierarchy. For constant depth circuits i.e., when $m=O(n)$, this linear scaling in th…
We develop a polynomial method on finite fields to amplify the hardness of spare sets in nondeterministic time complexity classes on a randomized streaming model. One of our results shows that if there exists a $2^{n^{o(1)}}$-sparse set in $NTIME(2^{n^{o(1)}})$ that does not have any randomized streaming algorithm with $n^{o(1)}$ updating time, and $n^{o(1)}$ space, then $NEXP\not=BPP$, where a $f(n)$-sparse set is a language that has at most $f(n)$ strings of length $n$. We also show that if MCSP is $ZPP$-hard under polynomial time truth-table reductions, then $EXP\not=ZPP$.
The Strong Exponential Time Hypothesis (SETH) asserts that for every $\varepsilon>0$ there exists $k$ such that $k$-SAT requires time $(2-\varepsilon)^n$. The field of fine-grained complexity has leveraged SETH to prove quite tight conditional lower bounds for dozens of problems in various domains and complexity classes, including Edit Distance, Graph Diameter, Hitting Set, Independent Set, and Orthogonal Vectors. Yet, it has been repeatedly asked in the literature whether SETH-hardness results can be proven for other fundamental problems such as Hamiltonian Path, Independent Set, Chromatic Nu…
In neuroscience, an important aspect of understanding the function of a neural circuit is to determine which, if any, of the neurons in the circuit are vital for the biological behavior governed by the neural circuit. A similar problem is to determine whether a given small set of neurons may be enough for the behavior to be displayed, even if all other neurons in the circuit are deactivated. Such a subset of neurons forms what is called a degenerate circuit for the behavior being studied. Recent advances in experimental techniques have provided researchers with tools to activate and deactivate…
Large-scale quantum information processing requires the use of quantum error correcting codes to mitigate the effects of noise in quantum devices. Topological error-correcting codes, such as surface codes, are promising candidates as they can be implemented using only local interactions in a two-dimensional array of physical qubits. Procedures such as defect braiding and lattice surgery can then be used to realize a fault-tolerant universal set of gates on the logical space of such topological codes. However, error correction also introduces a significant overhead in computation time, the numb…
Let $\mathcal{L}$ be a language that can be decided in linear space and let $ε>0$ be any constant. Let $\mathcal{A}$ be the exponential hardness assumption that for every $n$, membership in $\mathcal{L}$ for inputs of length~$n$ cannot be decided by circuits of size smaller than $2^{εn}$. We prove that for every function $f :\{0,1\}^* \rightarrow \{0,1\}$, computable by a randomized logspace algorithm $R$, there exists a deterministic logspace algorithm $D$ (attempting to compute $f$), such that on every input $x$ of length $n$, the algorithm $D$ outputs one of the following: 1: The correct va…
A hitting-set generator (HSG) is a polynomial map $G:\mathbb{F}^k \to \mathbb{F}^n$ such that for all $n$-variate polynomials $C$ of small enough circuit size and degree, if $C$ is nonzero, then $C\circ G$ is nonzero. In this paper, we give a new construction of such an HSG assuming that we have an explicit polynomial of sufficient hardness. Formally, we prove the following over any field of characteristic zero: Let $k\in \mathbb{N}$ and $δ> 0$ be arbitrary constants. Suppose $\{P_d\}_{d\in \mathbb{N}}$ is an explicit family of $k$-variate polynomials such that $\operatorname{deg} P_d = d$ and…
Noise is the defining feature of the NISQ era, but it remains unclear if noisy quantum devices are capable of quantum speedups. Quantum supremacy experiments have been a major step forward, but gaps remain between the theory behind these experiments and their actual implementations. In this work we initiate the study of the complexity of quantum random circuit sampling experiments with realistic amounts of noise. Actual quantum supremacy experiments have high levels of uncorrected noise and exponentially decaying fidelities. It is natural to ask if there is any signal of exponential complexity…
Quantum hardness of learning shallow classical circuits
Center for Theoretical Physics, MIT（麻省理工学院理论物理中心）；CWI（CWI数学与计算机科学中心）；QuSoft（QuSoft量子软件研究中心）；Joint Center for Quantum Information and Computer Science, University of Maryland（马里兰大学量子信息与计算机科学联合中心）
本文研究了均匀分布和分布无关的PAC学习框架下等深度经典电路的量子可学性。为了实现我们的结果，我们建立了量子学习和量子安全密码系统之间的联系。然后，我们实现以下结果。1)在均匀分布下学习AC $^0$和TC $^0$的难度。我们的第一个结果涉及概念类TC $^0$（resp. AC $^0$），具有无界扇入多数门（分别为）的恒定深度和多项式大小的电路类。AND、OR、NOT门）。我们证明，如果不存在量子多项式时间（分别）。强次指数时间）算法来解决带误差的环学习（RLWE）问题，那么TC $^0$（分别为. AC $^0$）（即使访问量子成员资格查询）。该结果中的主要技术使用被认为是量子安全的显式伪随机函数来构造在均匀分布下难以量子学习的概念类。2)在PAC设置中学习TC $^0_2$的难度。我们的第二个结果表明，如果不存在LWE问题的量子多项式时间算法，则不存在TC类$^0_2$的多项式时间量子PAC学习算法，即，深度2 TC $^0$电路。在这个结果中的主要技术是建立一个连接的公钥密码系统的量子安全性和可学习的概念类，包括解密功能的密码系统。这对Aaronson [Aar 05]提出的“量子计算理论的十大半重大挑战”之一给出了强（条件）否定的答案，Aaronson询问AC $^0$和TC $^0$是否可以在量子多项式时间内进行PAC学习。
We consider the problem of finding nearly optimal solutions of optimization problems with random objective functions. Two concrete problems we consider are (a) optimizing the Hamiltonian of a spherical or Ising $p$-spin glass model, and (b) finding a large independent set in a sparse Erdős-Rényi graph. The following families of algorithms are considered: (a) low-degree polynomials of the input; (b) low-depth Boolean circuits; (c) the Langevin dynamics algorithm. We show that these families of algorithms fail to produce nearly optimal solutions with high probability. For the case of Boolean cir…
The minimum circuit size problem (MCSP) is a string compression problem with a parameter $s$ in which, given the truth table of a Boolean function over inputs of length $n$, one must answer whether it can be computed by a Boolean circuit of size at most $s(n) \ge n$. Recently, McKay, Murray, and Williams (STOC, 2019) proved a hardness magnification result for MCSP involving (one-pass) streaming algorithms: For any reasonable $s$, if there is no $\mathsf{poly}(s(n))$-space streaming algorithm with $\mathsf{poly}(s(n))$ update time for $\mathsf{MCSP}[s]$, then $\mathsf{P} \neq \mathsf{NP}$. We p…
The ZX-calculus is a graphical language for reasoning about quantum computation using ZX-diagrams, a certain flexible generalisation of quantum circuits that can be used to represent linear maps from $m$ to $n$ qubits for any $m,n \geq 0$. Some applications for the ZX-calculus, such as quantum circuit optimisation and synthesis, rely on being able to efficiently translate a ZX-diagram back into a quantum circuit of comparable size. While several sufficient conditions are known for describing families of ZX-diagrams that can be efficiently transformed back into circuits, it has previously been …
We use the approximation method of Razborov to analyze the locality barrier which arose from the investigation of the hardness magnification approach to complexity lower bounds. Adapting a limitation of the approximation method obtained by Razborov, we show that in many cases it is not possible to combine the approximation method with typical (localizable) hardness magnification theorems to derive strong circuit lower bounds. In particular, one cannot use the approximation method to derive an extremely strong constant-depth circuit lower bound and then magnify it to an $NC^1$ lower bound for a…
In this work, we initiate the study of the Minimum Circuit Size Problem (MCSP) in the quantum setting. MCSP is a problem to compute the circuit complexity of Boolean functions. It is a fascinating problem in complexity theory -- its hardness is mysterious, and a better understanding of its hardness can have surprising implications to many fields in computer science. We first define and investigate the basic complexity-theoretic properties of minimum quantum circuit size problems for three natural objects: Boolean functions, unitaries, and quantum states. We show that these problems are not tri…
A hitting set construction, with application to arithmetic circuit lower bounds
多项式同一性测试算法必须确定给定的输入多项式是否完全等于0。给出了一元多项式$\sum_{j=0}^t c_j X^{α_j} (a + b X)^{β_j}$的确定性黑盒恒等式检验算法。从我们的算法中，我们推导出一个指数下界表示的多项式，如$\prod_{i=1}^{2^n} (X^i-1)$在这种形式下。已经证明，这些多项式很难用一般的算术电路来计算。我们的结果表明，“从去随机化的硬度”的方法是可行的，为一类受限制的算术电路的下界。证明是基于代数数论的技术，更确切地说，是基于代数数的高度函数的性质。
What makes normalized weighted satisfiability tractable
We consider the weighted antimonotone and the weighted monotone satisfiability problems on normalized circuits of depth at most $t \geq 2$ , abbreviated { \sc wsat $^-[t]$ } and { \sc wsat $^+[t]$ }, respectively. These problems model the weighted satisfiability of antimonotone and monotone propositional formulas (including weighted anitmonoone/monotone { \sc cnf-sat}) in a natural way, and serve as the canonical problems in the definition of the parameterized complexity hierarchy. We characterize the parameterized complexity of { \sc wsat $^-[t]$ } and { \sc wsat $^+[t]$ } with respect to the…
Evaluating QAOA expectation values can be as hard as counting optimal solutions
Quantum Artificial Intelligence Laboratory (QuAIL), NASA Ames Research Center（量子人工智能实验室（QuAIL），NASA艾姆斯研究中心）；USRA Research Institute for Advanced Computer Science（美国大学空间研究协会高级计算机科学研究院）
