John Preskill

High-rate quantum LDPC (qLDPC) codes reduce space overhead by densely packing many logical qubits into a single block of physical qubits. Here we extend such savings to computation by constructing batched fault-tolerant operations that apply the same logical gate across many code blocks in parallel. By leveraging shared physical resources to execute many logical operations in parallel, these operations realize high rates in space-time and significantly reduce computational costs. For arbitrary CSS qLDPC codes, we build batched gadgets with constant space-time overhead for (i) single-shot error correction and state preparation, (ii) code switching, and (iii) addressable Clifford gates. Using these batched gadgets we also construct parallel non-Clifford gates with low space-time cost. We outline principles for designing parallel quantum algorithms optimized for a batched architecture, and show in particular how lattice Hamiltonian dynamical simulations can be compiled efficiently. We also propose a near-term–friendly implementation using new self-dual Bivariate-Bicycle codes with high encoding rates (∼ 1/10), transversal Clifford gates, and global T gates, enabling Hamiltonian simulations with a lower space-time cost than analogous surface-code protocols and low-rate qLDPC protocols. These results open new paths toward scalable quantum computation via co-design of parallel quantum algorithms and high-rate fault-tolerant protocols.

We introduce transversal dimension jump, a code-switching protocol for lifted product (LP) quantum low-density parity-check (qLDPC) codes across different chain-complex dimensions, enabling universal fault-tolerant quantum computation with low overhead. The construction leverages the product structure of LP codes to implement one-way transversal CNOTs between a 3D code and its 2D component codes, enabling teleportation-based switching with geometrically nonlocal gates. Combined with constant-depth CCZ gates in 3D LP codes and low-overhead transversal Clifford gates in 2D LP codes, this yields universal, high-rate quantum logical computation with high thresholds and low space-time costs. Beyond asymptotic schemes, we identify explicit 3D–2D LP code pairs supporting cup-product CCZ gates, including bivariate tricycle–bicycle families such as the [[81, 3, 5]]–[[54, 2, 6]] pair, where the 3D tricycle codes admit depth-2 CCZ, weight-6 stabilizers, and pseudo-thresholds >~0.4%. As a byproduct, we show that the 3D codes enable highly efficient magic-state preparation: a single round of stabilizer measurements followed by depth-2 CCZ and postselection produces states with error <10^{-9} and success probability ~35%. Our results establish a native integration of qLDPC codes with complementary transversal gates—covering nearly all practically relevant families known so far—and open a broad design space for scalable, low-overhead universal quantum computation.

Constant-rate low-density parity-check (LDPC) codes are promising candidates for constructing efficient fault-tolerant quantum memories. However, if physical gates are subject to geometric-locality constraints, it becomes challenging to realize these codes. In this paper, we construct a new family of [[N,K,D]] codes, referred to as hierarchical codes, that encode a number of logical qubits K = Omega(N/łog(N)^2). The N-th element of this code family is obtained by concatenating a constant-rate quantum LDPC code with a surface code; nearest-neighbor gates in two dimensions are sufficient to implement the corresponding syndrome-extraction circuit and achieve a threshold. Below threshold the logical failure rate vanishes superpolynomially as a function of the distance D(N). We present a bilayer architecture for implementing the syndrome-extraction circuit, and estimate the logical failure rate for this architecture. Under conservative assumptions, we find that the hierarchical code outperforms the basic encoding where all logical qubits are encoded in the surface code.

Decoders that provide an estimate of the probability of a logical failure conditioned on the error syndrome (``soft-output decoders'') can reduce the overhead cost of fault-tolerant quantum memory and computation. In this work, we construct efficient soft-output decoders for the surface code derived from the Minimum-Weight Perfect Matching and Union-Find decoders. We show that soft-output decoding can improve the performance of a ``hierarchical code,'' a concatenated scheme in which the inner code is the surface code, and the outer code is a high-rate quantum low-density parity-check code. Alternatively, the soft-output decoding can improve the reliability of fault-tolerant circuit sampling by flagging those runs that should be discarded because the probability of a logical error is intolerably large.

Abstract We reconsider the black hole firewall puzzle, emphasizing that quantum error-correction, computational complexity, and pseudorandomness are crucial concepts for understanding the black hole interior. We assume that the Hawking radiation emitted by an old black hole is pseudorandom, meaning that it cannot be distinguished from a perfectly thermal state by any efficient quantum computation acting on the radiation alone. We then infer the existence of a subspace of the radiation system which we interpret as an encoding of the black hole interior. This encoded interior is entangled with the late outgoing Hawking quanta emitted by the old black hole, and is inaccessible to computationally bounded observers who are outside the black hole. Specifically, efficient operations acting on the radiation, those with quantum computational complexity polynomial in the entropy of the remaining black hole, commute with a complete set of logical operators acting on the encoded interior, up to corrections which are exponentially small in the entropy. Thus, under our pseudorandomness assumption, the black hole interior is well protected from exterior observers as long as the remaining black hole is macroscopic. On the other hand, if the radiation is not pseudorandom, an exterior observer may be able to create a firewall by applying a polynomial-time quantum computation to the radiation.

Abstract Machine learning (ML) provides the potential to solve challenging quantum many-body problems in physics and chemistry. Yet, this prospect has not been fully justified. In this work, we establish rigorous results to understand the power of classical ML and the potential for quantum advantage in an important example application: predicting outcomes of quantum mechanical processes. We prove that for achieving a small average prediction error, one can always design a classical ML model whose sample complexity is comparable to the best quantum ML model (up to a small polynomial factor). Regarding computational complexity, we show that the class of problems that can be solved by efficient classical ML models with access to sampled data is strictly larger than BPP. Hence, classical ML models may be able to solve some challenging quantum problems after training from data obtained in physical experiments. As a concrete example, we prove that a simple, classical ML model can efficiently learn to predict ground state representations that approximate expectation values of local observables up to a small, constant error. This holds for any smooth family of gapped local Hamiltonians in a finite spatial dimension.

Abstract Noise in quantum metrology reduces the sensitivity to which one can determine an unknown parameter in the evolution of a quantum state, such as time. Here, we consider a probe system prepared in a pure state that evolves according to a given Hamiltonian. We study the resulting local sensitivity of the probe to time after the application of a given noise channel. We show that the decrease in sensitivity due to the noise is equal to the sensitivity that the environment gains with respect to the energy of the probe. We obtain necessary and sufficient conditions for when the probe does not suffer any sensitivity loss; these conditions are analogous to, but weaker than, the Knill-Laflamme quantum error correction conditions. New upper bounds on the sensitivity of the noisy probe are obtained via our uncertainty relation, by applying known sensitivity lower bounds on the environments system. Our time-energy uncertainty relation also generalizes to any two arbitrary parameters whose evolutions are generated by Hermitian operators. This uncertainty relation asserts a general trade-off between the sensitivities that two parties can achieve for any two respective parameters of a single quantum system, in terms of the commutator of the associated generators. We consider applications to strongly interacting many-body probes. We find probe states for general interaction graphs of Ising and Heisenberg interactions that are robust to any single located error. For a 1D spin chain with nearest-neighbor interactions subject to amplitude damping noise on each site, we verify numerically that our probe state does not lose any sensitivity to first order in the noise parameter.

The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high energy physics. The quantum complexity of a unitary transformation or quantum state is defined as the size of the shortest quantum computation that executes the unitary or prepares the state. It is reasonable to expect that the complexity of a quantum state governed by a chaotic many-body Hamiltonian grows linearly with time for a time that is exponential in the system size; however, because it is hard to rule out a short-cut that improves the efficiency of a computation, it is notoriously difficult to derive lower bounds on quantum complexity for particular unitaries or states without making additional assumptions. To go further, one may study more generic models of complexity growth. We provide a rigorous connection between complexity growth and unitary k-designs, ensembles which capture the randomness of the unitary group. This connection allows us to leverage existing results about design growth to draw conclusions about the growth of complexity. We prove that local random quantum circuits generate unitary transformations whose complexity grows linearly for a long time, mirroring the behavior one expects in chaotic quantum systems and verifying conjectures by Brown and Susskind. Moreover, our results apply under a strong definition of quantum complexity based on optimal distinguishing measurements.

The energy cost of erasing quantum states depends on our knowledge of the states. We show that learning algorithms can acquire such knowledge to erase many copies of an unknown state at the optimal energy cost. This is proved by showing that learning can be made fully reversible and has no fundamental energy cost itself. With simple counting arguments, we relate the energy cost of erasing quantum states to their complexity, entanglement, and magic. We further show that the constructed erasure protocol is computationally efficient when learning is efficient. Conversely, under standard cryptographic assumptions, we prove that the optimal energy cost cannot be achieved efficiently in general. These results also enable efficient work extraction based on learning. Together, our results establish a concrete connection between quantum learning theory and thermodynamics, highlighting the physical significance of learning processes and enabling provably-efficient learning-based protocols for thermodynamic tasks.