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…
A critical milestone on the path to useful quantum computers is quantum supremacy - a demonstration of a quantum computation that is prohibitively hard for classical computers. A leading near-term candidate, put forth by the Google/UCSB team, is sampling from the probability distributions of randomly chosen quantum circuits, which we call Random Circuit Sampling (RCS). In this paper we study both the hardness and verification of RCS. While RCS was defined with experimental realization in mind, we show complexity theoretic evidence of hardness that is on par with the strongest theoretical propo…
We show that lower bounds for explicit constant-variate polynomials over fields of characteristic $p > 0$ are sufficient to derandomize polynomial identity testing over fields of characteristic $p$. In this setting, existing work on hardness-randomness tradeoffs for polynomial identity testing requires either the characteristic to be sufficiently large or the notion of hardness to be stronger than the standard syntactic notion of hardness used in algebraic complexity. Our results make no restriction on the characteristic of the field and use standard notions of hardness. We do this by combinin…
This paper aims to present an advanced version of PP-hardness proof of Minesweeper by Bondt. The advancement includes improved Minesweeper configurations for 'logic circuits' in a hexagonal Minesweeper. To do so, I demonstrate logical uncertainty in Minesweeper, which ironically allows a possibility to make some Boolean operators. The fact that existing hexagonal logic circuits did not clearly distinguish the true and false signal needs an improved form of a hexagonal wire. I introduce new forms of logic circuits such as NOT, AND, OR gates, a curve and a splitter of wires. Moreover, these new …
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…
Recently, Google announced the first demonstration of quantum computational supremacy with a programmable superconducting processor. Their demonstration is based on collecting samples from the output distribution of a noisy random quantum circuit, then applying a statistical test to those samples called Linear Cross-Entropy Benchmarking (Linear XEB). This raises a theoretical question: how hard is it for a classical computer to spoof the results of the Linear XEB test? In this short note, we adapt an analysis of Aaronson and Chen [2017] to prove a conditional hardness result for Linear XEB spo…
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…
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）
In this paper we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of PAC learning. In order to attain our results, we establish connections between quantum learning and quantum-secure cryptosystems. We then achieve the following results. 1) Hardness of learning AC$^0$ and TC$^0$ under the uniform distribution. Our first result concerns the concept class TC$^0$ (resp. AC$^0$), the class of constant-depth and polynomial-sized circuits with unbounded fan-in majority gates (resp. AND, OR, NOT gates). We…
Recent work of Bravyi et al. and follow-up work by Bene Watts et al. demonstrates a quantum advantage for shallow circuits: constant-depth quantum circuits can perform a task which constant-depth classical (i.e., AC$^0$) circuits cannot. Their results have the advantage that the quantum circuit is fairly practical, and their proofs are free of hardness assumptions (e.g., factoring is classically hard, etc.). Unfortunately, constant-depth classical circuits are too weak to yield a convincing real-world demonstration of quantum advantage. We attempt to hold on to the advantages of the above resu…
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…
Charles University in Prague（布拉格查理大学）；University of Oxford（牛津大学）
We prove several results giving new and stronger connections between learning, circuit lower bounds and pseudorandomness. Among other results, we show a generic learning speedup lemma, equivalences between various learning models in the exponential time and subexponential time regimes, a dichotomy between learning and pseudorandomness, consequences of non-trivial learning for circuit lower bounds, Karp-Lipton theorems for probabilistic exponential time, and NC$^1$-hardness for the Minimum Circuit Size Problem.
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…
Hardness magnification reduces major complexity separations (such as $\mathsf{\mathsf{EXP}} \nsubseteq \mathsf{NC}^1$) to proving lower bounds for some natural problem $Q$ against weak circuit models. Several recent works [OS18, MMW19, CT19, OPS19, CMMW19, Oli19, CJW19a] have established results of this form. In the most intriguing cases, the required lower bound is known for problems that appear to be significantly easier than $Q$, while $Q$ itself is susceptible to lower bounds but these are not yet sufficient for magnification. In this work, we provide more examples of this phenomenon, and …
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。
In a sequence of seminal results in the 80's, Kaltofen showed that the complexity class VP is closed under taking factors. A natural question in this context is to understand if other natural classes of multivariate polynomials, for instance, arithmetic formulas, algebraic branching programs, bounded depth arithmetic circuits or the class VNP, are closed under taking factors. In this paper, we show that all factors of degree at most $\log^a n$ of polynomials with poly(n) size depth $k$ circuits have poly(n) size circuits of depth at most $O(k + a)$. This partially answers a question of Shpilka…
We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of $\SL_n(\C)$-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions). We develop a general framework, denoted \emph{algebraic circuit search problems}, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some…
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…
We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted as MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions from supposedly-intractable problems to MCSP / MKTP hinged on the power of MCSP / MKTP to distinguish random distributions from distributions produced by hardness-based pseudorandom generator constructions. We develop a fundamentally different approach inspired by the well-known interactive p…
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）
