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arXiv:2605.20334

Halving the cost of QROM

Danial Motlagh, Matthew Pocrnic

quant-phSubmitted May 19, 2026• Announced May 21, 2026• 0 citations
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Abstract

Table lookup, often referred to as quantum read only memory (QROM), is one of the most widely used subroutines in quantum algorithms, and constitutes the majority share of algorithmic overheads in most practical applications of quantum computers. It involves the coherent loading of $N$ bitstrings of length $b$ in superposition, and naively has a non-Clifford cost of $N$ Toffolis. It is known that given access to $b\, λ$ dirty qubits, one can reduce the Toffoli cost of QROM to $2\frac{N}λ + 4b(λ- 1)$. In this work, we first present an optimization to reduce this cost to $2\frac{N}λ + 2b(λ- 1) + 2λ-6$ by replacing the ``SelectSwap" architecture with ``SelectCopy". We then provide a further optimization for the qubit-constrained regime where the Toffoli cost is typically $\sim 2\frac{N}λ$, and reduce it to $\sim (1+\frac{1}{b})\frac{N}λ$, cutting the cost by approximately $50\%$ and effectively matching the performance of clean-qubit QROM using dirty qubits for practical values of $b$. Lastly, we provide a parametric family of methods that allow the interpolation of the prefactor of the $\frac{N}λ $ term from $2$ to ($\, 1+\frac{1}{b}\,$) to obtain the best cost for different qubit availability regimes.

Comments
8 pages, 4 figures

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Table lookup, often referred to as quantum read only memory (QROM), is one of the most widely used subroutines in quantum algorithms, and constitutes the majority share of algorithmic overheads in most practical applications of quantum computers. It involves the coherent loading of NNN bitstrings of length bbb in superposition, and naively has a non-Clifford cost of NNN Toffolis. It is known that given access to b λb\, λbλ dirty qubits, one can reduce the Toffoli cost of QROM to 2Nλ+4b(λ−1)2\frac{N}λ + 4b(λ- 1)2λN​+4b(λ−1). In this work, we first present an optimization to reduce this cost to 2Nλ+2b(λ−1)+2λ−62\frac{N}λ + 2b(λ- 1) + 2λ-62λN​+2b(λ−1)+2λ−6 by replacing the SelectSwap" architecture with SelectCopy". We then provide a further optimization for the qubit-constrained regime where the Toffoli cost is typically ∼2Nλ\sim 2\frac{N}λ∼2λN​, and reduce it to ∼(1+1b)Nλ\sim (1+\frac{1}{b})\frac{N}λ∼(1+b1​)λN​, cutting the cost by approximately 50%50\%50% and effectively matching the performance of clean-qubit QROM using dirty qubits for practical values of bbb. Lastly, we provide a parametric family of methods that allow the interpolation of the prefactor of the Nλ\frac{N}λ λN​ term from 222 to ( 1+1b \, 1+\frac{1}{b}\,1+b1​) to obtain the best cost for different qubit availability regimes.
Quantum read only memory (QROM) is a substantial source of non-Clifford cost in fault-tolerant quantum algorithms. Borrowing dirty qubits can reduce this cost, provided their unknown initial states are restored. We consider the practically relevant qubit-constrained regime, where limited workspace restricts the table-block size and table loading dominates the cost. For NNN entries of bbb bits and approximately bλbλbλ dirty qubits, SelectSwap has leading Toffoli count 2N/λ2N/λ2N/λ. Recent work using sequential bit packets reduces this to (1+1/b)N/λ(1+1/b)N/λ(1+1/b)N/λ. In this work we introduce a dense encoding construction of QROM, where two bits of classical information are stored in each dirty qubit instead of one by utilizing both the ZZZ and XXX bases. On its own, dense encoding provides a 2×2\times2× improvement over SelectSwap by reducing the leading Toffoli count to N/λN/λN/λ. Combining it with sequential bit packets gives 12(1+1/b)N/λ\tfrac12(1+1/b)N/λ21​(1+1/b)N/λ, a further halving of the sequential construction's leading cost, and an approximately 4×4\times4× improvement over SelectSwap in the large-NNN limit for practical values of bbb.
Many quantum algorithms require coherent access to classical data, often modeled by quantum read-only memory (QROM). We initiate the study of the T\mathrm{T}T count of sparse QROM, in which only sss of the 2n2^n2n addresses store nonzero data. We prove that the optimal T\mathrm{T}T count is Θ(n+min⁡{s,s(m+log⁡(2n+1/s))})Θ\left(n+\min\left\{s,\sqrt{s\left(m+\log(2^{n+1}/s)\right)}\right\}\right)Θ(n+min{s,s(m+log(2n+1/s))​}). Our upper bounds use a multilevel hashing scheme, while our lower bounds reduce sparse QROM to state preparation and use counting arguments for adaptive Clifford+T\mathrm{T}T circuits. The lower bounds thus hold even when mid-circuit measurements and classically controlled operations are allowed. As applications, we obtain matching T\mathrm{T}T-count bounds Θ(min⁡{s,slog⁡(2n+1/s)}+slog⁡(1/ε)+log⁡(1/ε))Θ\left(\min\left\{s,\sqrt{s\log(2^{n+1}/s)}\right\} +\sqrt{s\log(1/\varepsilon)}+\log(1/\varepsilon)\right)Θ(min{s,slog(2n+1/s)​}+slog(1/ε)​+log(1/ε)) for sss-sparse state preparation and Θ(2ns(n+log⁡(1/εBE))+log⁡(1/εBE))Θ\left(\sqrt{2^n s\left(n+\log(1/\varepsilon_{\rm BE})\right)} +\log(1/\varepsilon_{\rm BE})\right)Θ(2ns(n+log(1/εBE​))​+log(1/εBE​)) for block encoding of sss-sparse matrices, where ε\varepsilonε and εBE\varepsilon_{\rm BE}εBE​ are the precision of state preparation and block encoding, respectively.
Quantum computing is a rapidly expanding field with applications ranging from optimization all the way to complex machine learning tasks. Quantum memories, while lacking in practical quantum computers, have the potential to bring quantum advantage. In quantum machine learning applications for example, a quantum memory can simplify the data loading process and potentially accelerate the learning task. Quantum memory can also store intermediate quantum state of qubits that can be reused for computation. However, the depth, gate count and compilation time of quantum memories such as, Quantum Read Only Memory (QROM) scale exponentially with the number of address lines making them impractical in state-of-the-art Noisy Intermediate-Scale Quantum (NISQ) computers beyond 4-bit addresses. In this paper, we propose techniques such as, pre-decoding logic and qubit reset to reduce the depth and gate count of QROM circuits to target wider address ranges such as, 8-bits. The proposed approach reduces the number of gates and depth count by at least 2X compared to the naive implementation at only 36% qubit overhead. A reduction in circuit depth and gate count as high as 75X and compilation time by 85X at the cost of a maximum of 2.28X qubit overhead is observed. Experimentally, the fidelity with the proposed pre-decoding circuit compared to existing optimization approach is also higher (as much as 73% compared to 40.8%) under reduced error rates.
We propose a programmable quantum read-only memory based on Gray code encoding, termed GQROM, designed to efficiently load classical data into quantum circuits. This architecture mitigates critical bottlenecks in near-term quantum computing by reducing initialization overhead and gate-induced errors. By utilizing Gray code, in which consecutive addresses differ by only a single bit, GQROM minimizes state transition complexity. We demonstrate the architecture's flexibility in accessing targeted data through initial-state modification. The entire design, including a systematic EDA flow, was implemented and validated on the Qiskit platform, confirming its practical feasibility. These results underscore GQROM's potential as a scalable and hardware-friendly solution for efficient state preparation in quantum circuits.
Quantum algorithms claim significant speedup over their classical counterparts for solving many problems. An important aspect of many of these algorithms is the existence of a quantum oracle, which needs to be implemented efficiently in order to realize the claimed advantages in practice. A quantum random access memory (QRAM) is a promising architecture for realizing these oracles. In this paper we develop a new design for QRAM and implement it with Clifford+T circuit. We focus on optimizing the T-count and T-depth since non-Clifford gates are the most expensive to implement fault-tolerantly in most error correction schemes. Integral to our design is a polynomial encoding of bit strings and so we refer to this design as [Formula: see text]. Compared to the previous state-of-the-art bucket brigade architecture for QRAM, we achieve an exponential improvement in T-depth, while reducing T-count and keeping the qubit-count same. Specifically, if N is the number of memory locations to be queried, then [Formula: see text] has T-depth [Formula: see text], T-count [Formula: see text] and uses O(N) logical qubits, while the bucket brigade circuit has T-depth [Formula: see text], T-count O(N) and uses O(N) qubits. Combining two [Formula: see text] we design a quantum look-up-table, [Formula: see text], that has T-depth [Formula: see text], T-count [Formula: see text] and qubit count [Formula: see text]. A quantum look-up table (qLUT) or quantum read-only memory (QROM) has restricted functionality than a QRAM. For example, it cannot write into a memory location and the circuit needs to be compiled each time the contents of the memory change. The previous state-of-the-art CSWAP architecture has T-depth [Formula: see text], T-count [Formula: see text] and qubit count [Formula: see text]. Thus we achieve a double exponential improvement in T-depth while keeping the T-count and qubit-count asymptotically same. Additionally, with our polynomial encoding of bit strings, we develop a method to optimize the Toffoli-count of circuits, specially those consisting of multi-controlled-NOT gates.
Quantum access to arbitrary classical data encoded in unitary black-box oracles underlies interesting data-intensive quantum algorithms, such as machine learning or electronic structure simulation. The feasibility of these applications depends crucially on gate-efficient implementations of these oracles, which are commonly some reversible versions of the boolean circuit for a classical lookup table. We present a general parametrized architecture for quantum circuits implementing a lookup table that encompasses all prior work in realizing a continuum of optimal trade-offs between qubits, non-Clifford gates, and error resilience, up to logarithmic factors. Our architecture assumes only local 2D connectivity, yet recovers results, with the appropriate parameters, polylogarithmic error scaling. We also identify regimes, such as simultaneous sublinear scaling, in all parameters. These results enable tailoring implementations of the commonly used lookup table primitive to any given quantum device with constrained resources.