A pseudorandom quantum state (PRS) is an ensemble of quantum states indistinguishable from Haar-random states to observers with efficient quantum computers. It allows one to substitute the costly Haar-random state with efficiently preparable PRS as a resource for cryptographic protocols, while also finding applications in quantum learning theory, black hole physics, many-body thermalization, quantum foundations, and quantum chaos. All existing constructions of PRS equate the notion of efficiency to quantum computers which runtime is bounded by a polynomial in its input size. In this work, we relax the notion of efficiency for PRS with respect to observers with near-term quantum computers implementing algorithms with runtime that scales slower than polynomial-time. We introduce the
T-PRS which is indistinguishable to quantum algorithms with runtime
T(n) that grows slower than polynomials in the input size
n. We give a set of reasonable conditions that a
T-PRS must satisfy and give two constructions by using quantum-secure pseudorandom functions and pseudorandom functions. For
T(n) being linearithmic, linear, polylogarithmic, and logarithmic function, we characterize the amount of quantum resources a
T-PRS must possess, particularly on its coherence, entanglement, and magic. Our quantum resource characterization applies generally to any two state ensembles that are indistinguishable to observers with computational power
T(n), giving a general necessary condition of whether a low-resource ensemble can mimic a high-resource ensemble, forming a
T-pseudoresource pair. We demonstate how the necessary amount of resource decreases as the observer's computational power is more restricted, giving a
T-pseudoresource pair with larger resource gap for more computationally limited observers.
Deep thermalization refers to the emergence of Haar-like randomness from quantum systems upon partial measurements. As a generalization of quantum thermalization, it is often associated with high complexity and entanglement. Here, we introduce computational deep thermalization and construct the fastest possible dynamics exhibiting it at infinite effective temperature. Our circuit dynamics produce quantum states with low entanglement in polylogarithmic depth that are indistinguishable from Haar random states to any computationally bounded observer. Importantly, the observer is allowed to request many copies of the same residual state obtained from partial projective measurements on the state-this condition is beyond the standard settings of quantum pseudorandomness but natural for deep thermalization. In cryptographic terms, these states are pseudorandom and pseudoentangled, and crucially, they retain these properties under local measurements. Our results demonstrate a new form of computational thermalization in which thermal-like behavior arises from structured quantum states endowed with cryptographic properties instead of from highly unstructured ensembles. The low resource complexity of preparing these states suggests scalable simulations of deep thermalization using quantum computers. Our Letter also motivates the study of computational quantum pseudorandomness beyond BQP observers.
A bstract The dynamics of quantum entanglement plays a central role in explaining the emergence of thermal equilibrium in isolated many-body systems. However, entanglement is notoriously hard to measure, and can in fact be “forged”: recent works have introduced a notion of pseudoentanglement describing ensembles of many-body states that, while only weakly entangled, cannot be efficiently distinguished from states with much higher entanglement, such as random states in the Hilbert space. This prompts the question: how much entanglement is truly necessary to achieve thermal equilibrium in quantum systems? In this work we address this question by introducing random circuit models of quantum dynamics that, at late times, equilibrate to pseudoentangled ensembles — a phenomenon we name ensemble pseudothermalization . These models replicate all the efficiently observable predictions of thermal equilibrium, while generating only a small (and tunable) amount of entanglement, thus deviating from the “maximum-entropy principle” that underpins thermodynamics. We examine (i) how a pseudoentangled ensemble on a small subsystem spreads to the whole system as a function of time, and (ii) how a pseudoentangled ensemble can be generated from an initial product state. We map the above problems onto a family of classical Markov chains on subsets of the computational basis. The mixing times of such Markov chains are related to the time scales at which the states produced from the dynamics become indistinguishable from Haar-random states at the level of each statistical moment, or number of copies. Based on a combination of rigorous bounds and conjectures supported by numerics, we argue that each Markov chain’s relaxation time and mixing time have different asymptotic behavior in the limit of large system size. This is a necessary condition for a cutoff phenomenon : an abrupt dynamical transition to equilibrium. We thus conjecture that our random circuits give rise to asymptotically sharp distinguishability transitions.
One of the state of the art techniques for classically simulating quantum circuits relies on approximating the output state of the circuit by a superposition of stabilizer states. \nIf the number of non-Clifford gates in the circuit is small, such simulations can be very effective. \nThis thesis provides various improvements in this framework. First, we describe an improved method of computing approximate stabilizer decompositions, which reduces the time cost of computing a single term in the decomposition from
O(elln2) to
O(mn2), where
ell is the total number of gates in the circuit, and
m is the number of non-Clifford gates. \nSince this subroutine has to be repeated exponentially many times, this improvement can be significant in practice whenever
ellggm. \nOur method uses a certain re-writing of the circuit, which in some cases allows for a significant amelioration of the exponential scaling of the required classical resources. \n \nFurthermore, we describe a method of constructing exact, low-rank stabilizer decompositions of
ketpsiotimesm, where
ketpsi is either a magic state or an equatorial state. \nFor any single qubit magic state
ketpsi, we find stabilizer decompositions of
ketpsiotimesm with
2mlog2(3)/4 terms. This improves on the best known bound of
2mlog2(7)/6. \nSimilarly, for any single qubit equatorial state
ketpsi, we give a stabilizer decomposition of
ketpsiotimesm with
2m/2 terms. \nTo our knowledge no such decompositions were previously known. \nThese results translate to milder exponential scaling of the classical resources required for estimating probabilities of quantum circuits up to a polynomially small multiplicative error, as well as allowing more types of circuits to be simulated in this way. \n \nWe also consider certain obstructions to classical simulations. \nIt has been argued in various contexts that contextuality and non-locality hamper classical simulations of quantum circuits. \nLinear constraint systems (LCSs) are a generalization of the well-known Peres-Mermin magic square, which has been recently used to prove a separation between the power of constant depth classical and quantum circuits. \n \nWhile binary LCSs have been studied in detail,
d-ary LCSs are less well-understood. In this thesis we consider linear constraint systems modulo
d>2. We give a simple proof, of the previously known fact, that any linear constraint system which admits a quantum solution consisting of generalized Pauli observables in odd dimension must be classically satisfiable. We further prove that, for odd
d, if a Pauli-like commutation relation between two variables in the LCS arises, then it has no quantum solutions in any dimensions, in stark contrast to the even
d case. We apply this result to various examples, for instance showing that many generalizations of the Peres-Mermin magic square do not give rise to a quantum vs. classical satisfiability gap.
The study of classical simulability is a key part of quantum information theory, providing new tools for fundamental research and the development of quantum computing. An increasingly successful framework for the study of simulability is resource theory. The best-known and best-understood quantum resource is entanglement, but in recent years nonstabilizerness, or magic, has also been gathering attention from the community. It is not yet well understood to what extent different resources play off each other in making systems hard to classically simulate. It has recently been shown that Clifford-augmented matrix product states (CAMPS) allow for efficient circuit simulation in some cases where both entanglement and magic are present. Another emerging method which has shown the ability to simulate systems with both resources is Pauli propagation (PP). Furthermore, the interplay between entanglement and magic at the level of physical phenomena is also poorly understood, in particular in the context of real-time dynamics. In this work, I implement an analytical algorithm proposed by Fux et al. (2025) which allows CAMPS to efficiently simulate essentially the first N non-Clifford Pauli exponential gates in an N-qubit quantum circuit, reproducing results obtained by greedy-search entanglement cooling in the paper. I then develop CAMPS-PP, a method for expectation value dynamics which evolves the state as a CAMPS and the observable in the Heisenberg picture using PP. I show that this method is significantly faster than pure CAMPS for slightly deep random Pauli rotation circuits at the tested system sizes, up to N=100, with improvements of up to two orders of magnitude for N=20. Finally, in order to investigate the physical relevance of magic I apply CAMPS-PP to an XXZ Heisenberg Floquet circuit with Clifford and non-Clifford angles, finding that CAMPS-PP slightly extends the simulable non-Clifford gate density and significantly reduces memory requirements compared to PP.
An ultimate goal of quantum computing is to perform calculations beyond the reach of any classical computer.It is therefore imperative that useful quantum computers be very difficult to simulate classically, otherwise classical computers could be used for the applications envisioned for the quantum ones.Perfect quantum computers are unarguably exponentially difficult to simulate: the classical resources required grow exponentially with the number of qubits N or the depth D of the circuit.This difficulty has triggered recent experiments on deep, random circuits that aim to demonstrate that quantum devices may already perform tasks beyond the reach of classical computing.These real quantum computing devices, however, suffer from many sources of decoherence and imprecision which limit the degree of entanglement that can actually be reached to a fraction of its theoretical maximum.They are characterized by an exponentially decaying fidelity F ∼ ð1 -ϵÞ ND with an error rate ϵ per operation as small as ≈1% for current devices with several dozen qubits or even smaller for smaller devices.In this work, we provide new insight on the computing capabilities of real quantum computers by demonstrating that they can be simulated at a tiny fraction of the cost that would be needed for a perfect quantum computer.Our algorithms compress the representations of quantum wave functions using matrix product states, which are able to capture states with low to moderate entanglement very accurately.This compression introduces a finite error rate ϵ so that the algorithms closely mimic the behavior of real quantum computing devices.The computing time of our algorithm increases only linearly with N and D in sharp contrast with exact simulation algorithms.We illustrate our algorithms with simulations of random circuits for qubits connected in both one-and twodimensional lattices.We find that ϵ can be decreased at a polynomial cost in computing power down to a minimum error ϵ ∞ .Getting below ϵ ∞ requires computing resources that increase exponentially with ϵ ∞ =ϵ.For a two-dimensional array of N ¼ 54 qubits and a circuit with control-Z gates, error rates better than state-of-the-art devices can be obtained on a laptop in a few hours.For more complex gates such as a SWAP gate followed by a controlled rotation, the error rate increases by a factor 3 for similar computing time.Our results suggest that, despite the high fidelity reached by quantum devices, only a tiny fraction ð∼10 -8 Þ of the system Hilbert space is actually being exploited.