The existence of quantum locally generated codes is a long standing open problem in quantum information theory. In this thesis, we consider a bound concerning this conjecture as well as a few constructions with codes that are `barely non-local'. We establish a complementary result to Bravyi and Terhal, and show quantum codes which are ``strongly'' not embeddable into finite dimensional lattices must also have poor distance. Along the way we derive some results concerning "pseudorandom" classical codes.\nGiven that quantum codes seem to be difficult to construct, it seems useful to examine "bad" quantum codes for applications in information and communication. Indeed, most of the work in quantum error correction by researchers today is in this direction. We construct a "nearly local" quantum erasure code which can achieve the capacity of the quantum erasure channel. This code has very poor (adversarial) distance, but still manages to correct random erasure errors with high probability. The codes use random Erdos-Renyi graphs to construct quantum states which are nearly local, but also highly entangled across fixed cuts with high probability. We derive some new results concerning classical codes with log-sparse parity check matrices which may be of independent interest.\nInspired by this construction, we are able to construct new approximate unitary 2-designs or "scramblers". The study of scrambling is the study of the mixing properties of different distributions of random unitaries. There is an inherent duality between the study of scrambling and the study of error correction: A good quantum code will make a good scrambler and vice versa. We study the scrambling properties of our random Erdos-Renyi graph state encoding circuits. We are able to show that these circuits, when supplemented by some local "Expander Graph" quantum circuits, form approximate unitary 2-designs. This construction, strictly speaking, does not achieve more efficient parameters than existing approximate 2-designs, but might have implementation advantages over other designs and points to a conjecture which could yield approximate unitary designs with time independent qubit-to-qubit coupling. This would be an extremely interesting construction in the context of experimental randomized benchmarking.
The existence of quantum locally generated codes is a long standing open problem in quantum information theory. In this thesis, we consider a bound concerning this conjecture as well as a few constructions with codes that are `barely non-local'. We establish a complementary result to Bravyi and Terhal, and show quantum codes which are ``strongly'' not embeddable into finite dimensional lattices must also have poor distance. Along the way we derive some results concerning "pseudorandom" classical codes.\nGiven that quantum codes seem to be difficult to construct, it seems useful to examine "bad" quantum codes for applications in information and communication. Indeed, most of the work in quantum error correction by researchers today is in this direction. We construct a "nearly local" quantum erasure code which can achieve the capacity of the quantum erasure channel. This code has very poor (adversarial) distance, but still manages to correct random erasure errors with high probability. The codes use random Erdos-Renyi graphs to construct quantum states which are nearly local, but also highly entangled across fixed cuts with high probability. We derive some new results concerning classical codes with log-sparse parity check matrices which may be of independent interest.\nInspired by this construction, we are able to construct new approximate unitary 2-designs or "scramblers". The study of scrambling is the study of the mixing properties of different distributions of random unitaries. There is an inherent duality between the study of scrambling and the study of error correction: A good quantum code will make a good scrambler and vice versa. We study the scrambling properties of our random Erdos-Renyi graph state encoding circuits. We are able to show that these circuits, when supplemented by some local "Expander Graph" quantum circuits, form approximate unitary 2-designs. This construction, strictly speaking, does not achieve more efficient parameters than existing approximate 2-designs, but might have implementation advantages over other designs and points to a conjecture which could yield approximate unitary designs with time independent qubit-to-qubit coupling. This would be an extremely interesting construction in the context of experimental randomized benchmarking.
This thesis is focused on the generation and understanding of particular kinds of quantum randomness. Randomness is useful for many tasks in physics and information processing, from randomized benchmarking , to black hole physics , as well demonstrating a so-called quantum speedup , and many other applications. On the one hand we explore how to generate a particular form of random evolution known as a t-design. On the other we show how this can also give instances for quantum speedup - where classical computers cannot simulate the randomness efficiently. We also show that this is still possible in noisy realistic settings. More specifically, this thesis is centered around three main topics. The first of these being the generation of epsilon-approximate unitary t-designs. In this direction, we first show that non-adaptive, fixed measurements on a graph state composed of poly(n,t,log(1/epsilon)) qubits, and with a regular structure (that of a brickwork state) effectively give rise to a random unitary ensemble which is a epsilon-approximate t-design. This work is presented in Chapter 3. Before this work, it was known that non-adaptive fixed XY measurements on a graph state give rise to unitary t-designs , however the graph states used there were of complicated structure and were therefore not natural candidates for measurement based quantum computing (MBQC), and the circuits to make them were complicated. The novelty in our work is showing that t-designs can be generated by fixed, non-adaptive measurements on graph states whose underlying graphs are regular 2D lattices. These graph states are universal resources for MBQC. Therefore, our result allows the natural integration of unitary t-designs, which provide a notion of quantum pseudorandomness which is very useful in quantum algorithms, into quantum algorithms running in MBQC. Moreover, in the circuit picture this construction for t-designs may be viewed as a constant depth quantum circuit, albeit with a polynomial number of ancillas. We then provide new constructions of epsilon-approximate unitary t-designs both in the circuit model and in MBQC which are based on a relaxation of technical requirements in previous constructions. These constructions are found in Chapters 4 and 5.
The efficiency of locally generating unitary designs, which capture statistical notions of quantum pseudorandomness, lies at the heart of wide-ranging areas in physics and quantum information technologies. While there are extensive potent methods and results for this problem, the evidently important setting where continuous symmetries or conservation laws (most notably U(1) and SU(d)) are involved is known to present fundamental difficulties. In particular, even the basic question of whether any local symmetric circuit can generate 2-designs efficiently (in time that grows at most polynomially in the system size) remains open with no circuit constructions provably known to do so, despite intensive efforts. In this work, we resolve this long-standing open problem for both U(1) and SU(d) symmetries by explicitly constructing local symmetric quantum circuits which we prove to converge to symmetric unitary 2-designs in polynomial time using a combination of representation theory, graph theory, and Markov chain methods. As a direct application, our constructions can be used to efficiently generate near-optimal covariant quantum error-correcting codes, confirming a conjecture in [PRX Quantum 3, 020314 (2022)].
Ground states of local Hamiltonians can be generally highly entangled: any quantum circuit that generates them, even approximately, must be sufficiently deep to allow coupling (entanglement) between any pair of qubits. Until now this property was not known to be “robust” - the marginals of such states to a subset of the qubits containing all but a small constant fraction of them may be only locally entangled, and hence approximable by shallow quantum circuits. In this work we construct a family of 16-local Hamiltonians for which any marginal of a ground state to a fraction at least 1-10-8of the qubits must be globally entangled. This provides evidence that quantum entanglement is not very fragile, and perhaps our intuition about its instability is an artifact of considering local Hamiltonians which are not only local but spatially local. Formally, it provides positive evidence for two wide-open conjectures in condensed-matter physics and quantum complexity theory which are the qLDPC conjecture, positing the existence of “good” quantum LDPC codes, and the NLTS conjecture [1] positing the existence of local Hamiltonians in which any low-energy state is highly entangled. Our Hamiltonian is based on applying the hypergraph product by Tillich-Zemor [2] to the repetition code with checks from an expander graph. A key tool in our proof is a new lower bound on the vertex expansion of the output of low-depth quantum circuits, which may be of independent interest.
Pseudorandom error-correcting codes (PRCs), introduced by Christ and Gunn [CRYPTO 2024], are classical error-correcting codes whose codewords are computationally indistinguishable from uniformly random strings. We initiate the study of quantum analogues of PRCs. We define quantum pseudorandom error-correcting codes (QPRCs). Assuming that Learning Parity with Noise (LPN) is hard for 2O(n)-time quantum algorithms, we construct QPRCs whose encodings are indistinguishable from Haar-random isometries. We call these pseudorandom isometric error-correcting codes (PRICs), and our construction is robust to all o(nlognloglogn)-local quantum noise, where n is the number of physical qubits. Under the same assumption, we also construct QPRCs whose encodings are indistinguishable from the completely depolarizing channel and that are robust to all αn-local quantum noise for some constant α>0. The latter QPRCs give a direct quantum analogue of classical PRCs.
To construct PRICs, we develop two technical ingredients of independent interest. First, we introduce a new classical cryptographic primitive which we call pseudorandom functional error-correcting codes (PRFCs) and construct them under the same LPN assumption. Second, we develop a new efficient decoding procedure within the codeword-stabilized (CWS) framework introduced in [Cross, Smith, Smolin, and Zeng, IEEE ISIT 2008], which is a generic framework for constructing quantum error-correcting codes by combining (possibly nonlinear) classical error-correcting codes and graphs. This resolves the open problem of finding a general method for efficiently decoding such CWS codes based on nonlinear classical codes, raised in [Li, Dumer, Grassl, and Pryadko, Physical Review A 2010].
The existence of quantum LDPC codes with minimal distance scaling linearly in the number of qubits is a central open problem in quantum information. Despite years of research good quantum LDPC codes are not known to exist, but at the very least it is known they cannot be defined on very regular topologies, like low-dimensional grids. In this work we establish a complementary result, showing that good quantum CSS codes which are sparsely generated require “structure” in the local terms that constrain the code space so as not to be “too-random” in a well-defined sense. To show this, we prove a weak converse to a theorem of Krasikov and Litsyn on weight distributions of classical codes due to which may be of independent interest: subspaces for which the distribution of weights in the dual space is approximately binomial have very few codewords of low weight, tantamount to having a non-negligible “approximate” minimal distance. While they may not have a large minimal non-zero weight, they still have very few words of low Hamming weight.
Unitary designs provide finite-moment approximations to Haar-random unitaries, with wide-ranging applications across physics and quantum information, from scrambling and black-hole dynamics to foundational primitives in quantum algorithms. Strong unitary designs capture a more demanding operational notion of approximation, requiring indistinguishability from Haar randomness even for quantum algorithms that may access a unitary not only in the forward direction, but also through its inverse, transpose, and complex conjugate. Motivated by the physical requirement that scrambling arise within the system itself, Schuster, Ma, Lombardi, Brandão, and Huang (arXiv:2509.26310) left open whether strong unitary designs can be generated in logarithmic depth using only the system qubits. For every fixed design order k and measurable-error tolerance, we construct strong approximate unitary k-designs in optimal Θ(logn) all-to-all circuit depth using only the n original system qubits. Our new ingredient is a logarithmic-depth Pauli-mixing bound for the perfect-matching ensemble, whose layers pair the qubits uniformly at random and apply independent random two-qubit gates. This bound controls the mixed forward-reverse two-query case, which we combine with existing design and gluing results to obtain strong unitary designs of arbitrary fixed order.