Andras Gilyen

We present a protocol for fault-tolerantly implementing the logical quantum random access memory (QRAM) operation, given access to a specialized, noisy QRAM device. For coherently accessing classical memories of size 2^n, our protocol consumes only poly(n) fault-tolerant quantum resources (logical gates, logical qubits, quantum error correction cycles, etc.), avoiding the need to perform active error correction on all Ω(2^n) components of the QRAM device. This is the first rigorous conceptual demonstration that a specialized, noisy QRAM device could be useful for implementing a fault-tolerant quantum algorithm. In fact, the fidelity of the device can be as low as 1/poly(n). The protocol queries the noisy QRAM device poly(n) times to prepare a sequence of n-qubit QRAM resource states, which are moved to a general-purpose poly(n)-size processor to be encoded into a QEC code, distilled, and fault-tolerantly teleported into the computation. To aid this protocol, we develop a new gate-efficient streaming version of quantum purity amplification that matches the optimal sample complexity in a wide range of parameters and is therefore of independent interest. The exponential reduction in fault-tolerant quantum resources comes at the expense of an exponential quantity of purely classical complexity---each of the n iterations of the protocol requires adaptively updating the 2^n-size classical dataset and providing the noisy QRAM device with access to the updated dataset at the next iteration. We show that this classical operation can be parallelized to poly(n) classical circuit depth, but only in a model where classical sparse matrix-vector multiplication for 2^n-dimensional vectors can be as well. While our protocol demonstrates that QRAM is more compatible with fault-tolerant quantum computation than previously thought, the need for significant classical computational complexity exposes potentially fundamental limitations to realizing a truly poly(n)-cost fault-tolerant QRAM.

Markov Chain Monte Carlo (MCMC) methods are an essential tool in classical algorithms design. Especially, the Metropolis sampling algorithm and Glauber dynamics have drastically advanced our understanding of material properties, reaction dynamics, phase transitions, and thermodynamics. Recently, there has been a new wave of quantum MCMC algorithms that draws inspiration from the cooling process in Nature to design continuous-time Quantum Markov chains (i.e., Lindbladians) satisfying (approximate) detailed balance. Nevertheless, the quantum analog of detailed balance, which has been central to classical Markov chain design and analysis, has posed a challenge to quantum algorithms design and has only recently been achieved exactly and (quasi)-locally for an efficiently implementable Lindbladian by [CKG23]. The construction of [CKG23] provably leads to an efficient Gibbs state preparation method in the high-temperature regime. However, proving fast mixing for low temperatures remains an open problem, apart from some (almost) integrable systems. Here we introduce (i) a new continuous-time Lindbladian construction that also leads to quasi-local and detailed-balanced dynamics, and (ii) show that it is fast mixing for high-temperature lattice Hamiltonians. The new construction's major advantage is that it does not increase the number of Kraus operators, which is particularly helpful for numerical studies. We exploit the resulting low Kraus rank through a (iii) novel custom variant of density matrix renormalization group (DMRG) for superoperators to provide numerical evidence for various 1D models (Transverse-field Ising, Heisenberg XXZ) that the Gibbs sampler is mixing fast. We also introduce (iv) new detailed-balanced discrete-time quantum channel variants of all existing continuous-time detailed-balanced Lindbladian construction and (v) show that they are also mixing fast at high-temperatures, and provide some preliminary (vi) resource estimates for their implementation confirming their algorithmic efficiency.

Abstract Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and quantum query complexities. We also show that, remarkably, permutation groups constructed out of these symmetries are essentially the only permutation groups that prevent super-polynomial quantum speedups. We prove this by fully characterizing the primitive permutation groups that allow super-polynomial quantum speedups. In contrast, in the adjacency list model for bounded-degree graphswhere graph symmetry is manifested differentlywe exhibit a property testing problem that shows an exponential quantum speedup. These results resolve open questions posed by Ambainis, Childs, and Liu (2010) and Montanaro and de Wolf (2013).

Abstract We demonstrate the possibility of (sub)exponential quantum speedup via a quantum algorithm that follows an adiabatic path of a gapped sparse Hamiltonian with no sign problem. This is in sharp contrast with frustration-free stoquastic Hamiltonians, where no such speedup is possible as shown by Bravyi and Terhal (2008). The Hamiltonian that exhibits this speed-up comes from the adjacency matrix of an undirected graph, and we can view the adiabatic evolution as an efficient $\mathcal{O}(\mathrm{poly}(n))$-time quantum algorithm for finding a specific "EXIT" vertex in the graph given the "ENTRANCE" vertex. On the other hand we show that if the graph is given via an adjacency-list oracle, there is no classical algorithm that finds the "EXIT" with probability greater than $\exp(-n^\delta)$ using at most $\exp(n^\delta)$ queries for $\delta= \frac15 - o(1)$. Our construction of the graph is somewhat similar to the ``welded-trees'' construction of Childs et al. (2003), but uses additional ideas for simultaneously achieving a spectral gap and a short adiabatic path.

Abstract Recently Chen and Gao~\cite{ChenGao2017} proposed a new quantum algorithm for Boolean polynomial system solving, motivated by the cryptanalysis of some post-quantum cryptosystems. The key idea of their approach is to apply a Quantum Linear System (QLS) algorithm to a Macaulay linear system over $\CC$, which is derived from the Boolean polynomial system. The efficiency of their algorithm depends on the condition number of the Macaulay matrix. In this paper, we give a strong lower bound on the condition number as a function of the Hamming weight of the solution. We describe a Grover-based exhaustive search algorithm that always outperforms their algorithm. Then, we improve upon Chen and Gao's algorithm by introducing the Boolean Macaulay linear system over $\CC$ by reducing the original Macaulay linear system. This improved algorithm could potentially significantly outperform the brute-force algorithm, when the Hamming weight of the solution is logarithmic in the number of variables. Furthermore, we provide a simple and more elementary proof of correctness for our improved algorithm using a reduction employing the Valiant-Vazirani affine hashing method, and also extend the result to polynomial systems over $\FF_q$ improving on subsequent work by Chen, Gao and Yuan \cite{ChenGao2018}. We also suggest a new approach for extracting the solution of the Boolean polynomial system via a generalization of the quantum coupon collector problem \cite{arunachalam2020quantum}.