We present a classical algorithm that, for any 3D geometrically-local, constant-depth quantum circuit $C$, and any bit string $x \in \{0,1\}^n$, can compute the quantity $| |^2$ to within any inverse-polynomial additive error in quasi-polynomial time. It is known that it is $\#P$-hard to compute this same quantity to within $2^{-n^2}$ additive error [Mov20]. The previous best known algorithm for this problem used $O(2^{n^{1/3}}\text{poly}(1/\epsilon))$ time to compute probabilities to within additive error $\epsilon$ [BGM20]. Notably, the [BGM20] paper included an elegant polynomial time algorithm for the same estimation task with 2D circuits, which makes a novel use of 1D Matrix Product States carefully tailored to the 2D geometry of the circuit in question. Surprisingly, it is not clear that it is possible to extend this use of MPS to address the case of 3D circuits in polynomial time. This raises a natural question as to whether the computational complexity of the 3D problem might be drastically higher than that of the 2D problem. In this work we address this question by exhibiting a quasi-polynomial time algorithm for the 3D case. In order to surpass the technical barriers encountered by previously known techniques we are forced to pursue a novel approach:
Our algorithm has a Divide-and-Conquer structure, constructing a recursive sub-division of the given 3D circuit using carefully designed block-encodings, each creating a 3D-local circuit on at most half the number of qubits as the original. This division step is then applied recursively, expressing the original quantity as a weighted sum of smaller and smaller 3D-local quantum circuits. A central technical challenge is to control correlations arising from the entanglement that may exist between the different circuit pieces produced this way.
We present a classical algorithm that, for any 3D geometrically-local, constant-depth quantum circuit C, and any bit string x∈{0,1}n, can compute the quantity ∣∣2 to within any inverse-polynomial additive error in quasi-polynomial time. It is known that it is #P-hard to compute this same quantity to within 2−n2 additive error [Mov20]. The previous best known algorithm for this problem used O(2n1/3poly(1/ϵ)) time to compute probabilities to within additive error ϵ [BGM20]. Notably, the [BGM20] paper included an elegant polynomial time algorithm for the same estimation task with 2D circuits, which makes a novel use of 1D Matrix Product States carefully tailored to the 2D geometry of the circuit in question. Surprisingly, it is not clear that it is possible to extend this use of MPS to address the case of 3D circuits in polynomial time. This raises a natural question as to whether the computational complexity of the 3D problem might be drastically higher than that of the 2D problem. In this work we address this question by exhibiting a quasi-polynomial time algorithm for the 3D case. In order to surpass the technical barriers encountered by previously known techniques we are forced to pursue a novel approach:
Our algorithm has a Divide-and-Conquer structure, constructing a recursive sub-division of the given 3D circuit using carefully designed block-encodings, each creating a 3D-local circuit on at most half the number of qubits as the original. This division step is then applied recursively, expressing the original quantity as a weighted sum of smaller and smaller 3D-local quantum circuits. A central technical challenge is to control correlations arising from the entanglement that may exist between the different circuit pieces produced this way.
We present a classical algorithm that, for any 3D geometrically-local, polylogarithmic-depth quantum circuitC, and any bit stringx∈{0,1}n, can compute the quantity∣⟨x∣C∣0⊗n⟩∣2to within any inverse-polynomial additive error in quasi-polynomial time. It is known that it is#P-hard to compute this same quantity to within2−n2additive error [1], [2], and worst-case hardness results for this task date back to [3]. The previous best known algorithm for this problem usedO(2n1/3poly(1/ϵ))time to compute probabilities to within additive errorϵ[4]. Notably, that paper, [4], included an elegant polynomial time algorithm for the same estimation task with 2D circuits, which makes a novel use of 1D Matrix Product States carefully tailored to the 2D geometry of the circuit in question. Surprisingly, it is not clear that it is possible to extend this use of MPS to address the case of 3D circuits in polynomial time. This raises a natural question as to whether the computational complexity of the 3D problem might be drastically higher than that of the 2D problem. In this work we address this question by exhibiting a quasi-polynomial time algorithm for the 3D case. In order to surpass the technical barriers encountered by previously known techniques we are forced to pursue a novel approach: constructing a recursive sub-division of the given 3D circuit using carefully designed block-encodings. To our knowledge this is the first use of the block-encoding technique in a purely classical algorithm. Our algorithm has a Divide-and-Conquer structure, demonstrating how to approximate the desired quantity via several instantiations of the same problem type, each involving 3D-local circuits on about half the number of qubits as the original. This division step is then applied recursively, expressing the original quantity as a weighted sum of smaller and smaller 3D-local quantum circuits. A central technical challenge is to control correlations arising from entanglement that may exist between the different circuit “pieces” produced this way. We believe that the division step, which makes a novel use of block-encodings [5], together with an Inclusion-Exclusion argument to reduce error in each recursive approximation, may be of independent interest.11The full version of this paper is available at arXiv:2012.05460. We refer the reader to the full text for proofs of all Lemmas and Theorems.
We give a deterministic classical algorithm that estimates ∣⟨x∣U∣0n⟩∣2 to additive error ε in poly(n,1/ε) time, where U is a constant-depth quantum circuit comprised of gates with bounded fan-in and arbitrary connectivity, and x is an arbitrary n-bit output string. This improves over prior state-of-the-art algorithms that takes nO(log(n)) time for the same task, nO(log(log(n)) when U is geometrically local, and nO(1) for 2D geometrically-local circuits.
We construct a family of 2D-local constant-depth quantum circuits that output states whose entanglement entropy across a specified cut cannot be estimated in quantum polynomial time. As constant-depth quantum circuits can be learned from polynomially many quantum samples, our resulting pseudoentangled states are implicitly public-key and not pseudorandom. This separates pseudoentanglement from pseudorandomness in the shallow-circuit regime: the former is possible, while the latter is not. The construction is based on the quantum intractability of the Dense-Sparse Learning Parity with Noise problem introduced in [DJ25] and uses a bounded-fan-in, bounded-fan-out classical randomized encoding for linear maps x↦Mx, which could be of independent interest. As applications, we obtain quantum hardness for the problem of learning the entanglement structure (across a fixed cut) of the ground-state of 1D and 2D local Hamiltonians. The 1D Hamiltonian has an inverse polynomial gap, whereas the 2D one has a constant gap. This complements the result of [BZZ24] that showed only factoring-based hardness for the 1D case, though achieving a volume versus area entanglement difference.
We present a classical algorithm that, for any D-dimensional geometrically-local, quantum circuit C of polylogarithmic-depth, and any bit string x∈0,1n, can compute the quantity ∣∣2 to within any inverse-polynomial additive error in quasi-polynomial time, for any fixed dimension D. This is an extension of the result [CC21], which originally proved this result for D=3. To see why this is interesting, note that, while the D=1 case of this result follows from standard use of Matrix Product States, known for decades, the D=2 case required novel and interesting techniques introduced in [BGM19]. Extending to the case D=3 was even more laborious and required further new techniques introduced in [CC21]. Our work here shows that, while handling each new dimension has historically required a new insight, and fixed algorithmic primitive, based on known techniques for D≤3, we can now handle any fixed dimension D>3. Our algorithm uses the Divide-and-Conquer framework of [CC21] to approximate the desired quantity via several instantiations of the same problem type, each involving D-dimensional circuits on about half the number of qubits as the original. This division step is then applied recursively, until the width of the recursively decomposed circuits in the Dth dimension is so small that they can effectively be regarded as (D−1)-dimensional problems by absorbing the small width in the Dth dimension into the qudit structure at the cost of a moderate increase in runtime. The main technical challenge lies in ensuring that the more involved portions of the recursive circuit decomposition and error analysis from [CC21] still hold in higher dimensions, which requires small modifications to the analysis in some places.
In this paper, we study the quantum learnability of constant-depth classical circuits under the uniform distribution and in the distribution-independent framework of probably approximately correct (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 PAC learning AC0 and TC0 under the uniform distribution. Our first result concerns the concept class TC0 (resp., AC0), the class of constant-depth, polynomial-sized circuits with unbounded fan-in majority gates (resp., AND,OR,NOT gates). We show the following: if there exists no quantum (quasi-)polynomial-time algorithm to solve the ring-learning with errors (RLWE) problem, then there exists no (quasi-)polynomial-time quantum learning algorithm for TC0; and if there exists no 2O(d1/η)-time quantum algorithm to solve RLWE with dimension d=O(polylogn) (for every constant η>2), then there exists no O(nlogνn)-time quantum learning algorithm for poly(n)-sized AC0 circuits (for a constant ν>0), matching the classical upper bound of Linial, Mansour and Nisan [J. ACM, 40 (1993), pp. 607--620], where the learning algorithms are under the uniform distribution (even with access to quantum membership queries). The main technique in these results uses an explicit family of pseudorandom functions that are believed to be quantum-secure to construct concept classes that are hard to learn quantumly under the uniform distribution. 2. Hardness of learning TC20 in the PAC setting. Our second result shows that if there exists no quantum polynomial-time algorithm for the LWE problem, then there exists no polynomial-time quantum-PAC learning algorithm for the class TC20, i.e., depth-2 TC0 circuits. The main technique in this result is to establish a connection between the quantum security of public-key encryption schemes and the learnability of a concept class that consists of decryption functions of the cryptosystem. Our results show that quantum resources do not give an exponential improvement to learning constant-depth polynomial-sized neural networks. This also gives a strong (conditional) negative answer to one of the “Ten Semi-Grand Challenges for Quantum Computing Theory" raised by Aaronson https://www.scottaaronson.com/writings/qchallenge.html, 2005.
Motivated by the recent experimental demonstrations of quantum supremacy, proving the hardness of the output of random quantum circuits is an imperative near term goal. We prove under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to withinexp(−Ω(mlogm))additive error is hard for any classical computer, wheremis the number of gates in the quantum computation. More precisely, we show that the above problem is #P-hard under BPPNPreduction. In the recent experiments, the quantum circuit has n-qubits and the architecture is a two-dimensional grid of sizen×n[1]. Indeed for constant depth circuits approximating the output probabilities to within2−Ω(nlogn)is hard. For circuits of depthlognornfor which the anti-concentration property holds, approximating the output probabilities to within2−Ω(nlog2n)and2−Ω(n3/2logn)is hard respectively. We then show that the hardness results extend to any open neighborhood of an arbitrary (fixed) circuit including the trivial circuit with identity gates. We made an effort to find the best proofs and proved these results from first principles, which do not use the standard techniques such as the Berlekamp–Welch algorithm, the usual Paturi's lemma, and Rakhmanov's result.