arXiv subjects
Publications and source records attributed to Michele Mosca.
We study the problem of exact synthesis for the Clifford+R gate set and give the explicit structure of the underlying Bruhat-Tits building for this group. In this process, we also give an alternative proof of the arithmetic nature of this gate set.
We present a complete set of rewrite rules for n-qutrit Clifford circuits where n is any non-negative integer. This is the first completeness result for any fragment of quantum circuits in odd prime dimensions. We first generalize Selinger's normal form for n-qubit Clifford circuits to the qutrit setting. Then, we present a rewrite system by which any Clifford circuit can be reduced to this normal form. We then simplify the rewrite rules in this procedure to a small natural set of rules, giving a clean presentation of the group of qutrit Clifford unitaries in terms of generators and relations.
Clustering algorithms are at the basis of several technological applications, and are fueling the development of rapidly evolving fields such as machine learning. In the recent past, however, it has become apparent that they face challenges stemming from datasets that span more spatial dimensions. In fact, the best-performing clustering algorithms scale linearly in the number of points, but quadratically with respect to the local density of points. In this work, we introduce qLUE, a quantum clustering algorithm that scales linearly in both the number of points and their density. qLUE is inspired by CLUE, an algorithm developed to address the challenging time and memory budgets of Event Reconstruction (ER) in future High-Energy Physics experiments. As such, qLUE marries decades of development with the quadratic speedup provided by quantum computers. We numerically test qLUE in several scenarios, demonstrating its effectiveness and proving it to be a promising route to handle complex data analysis tasks -- especially in high-dimensional datasets with high densities of points.
We introduce an improved CNOT synthesis algorithm that considers nearest-neighbour interactions and CNOT gate error rates in noisy intermediate-scale quantum (NISQ) hardware. Compared to IBM's Qiskit compiler, it improves the fidelity of a synthesized CNOT circuit by about 2 times on average (up to 9 times). It lowers the synthesized CNOT count by a factor of 13 on average (up to a factor of 162). Our contribution is twofold. First, we define a $\textsf{Cost}$ function by approximating the average gate fidelity $F_{avg}$. According to the simulation results, $\textsf{Cost}$ fits the error probability of a noisy CNOT circuit, $\textsf{Prob} = 1 - F_{avg}$, much tighter than the commonly used cost functions. On IBM's fake Nairobi backend, it matches $\textsf{Prob}$ to within $10^{-3}$. On other backends, it fits $\textsf{Prob}$ to within $10^{-1}$. $\textsf{Cost}$ accurately quantifies the dynamic error characteristics and shows remarkable scalability. Second, we propose a noise-aware CNOT routing algorithm, NAPermRowCol, by adapting the leading Steiner-tree-based connectivity-aware CNOT synthesis algorithms. A weighted edge is used to encode a CNOT gate error rate and $\textsf{Cost}$-instructed heuristics are applied to each reduction step. NAPermRowCol does not use ancillary qubits and is not restricted to certain initial qubit maps. Compared with algorithms that are noise-agnostic, it improves the fidelity of a synthesized CNOT circuit across varied NISQ hardware. Depending on the benchmark circuit and the IBM backend selected, it lowers the synthesized CNOT count up to $56.95\%$ compared to ROWCOL and up to $21.62\%$ compared to PermRowCol. It reduces the synthesis $\textsf{Cost}$ up to $25.71\%$ compared to ROWCOL and up to $9.12\%$ compared to PermRowCol. Our method can be extended to route a more general quantum circuit, giving a powerful new tool for compiling on NISQ devices.
Quantum computing is an emerging technology with potentially far-reaching implications for national prosperity and security. Understanding the timeframes over which economic benefits and national security risks may manifest themselves is vital for ensuring the prudent development of this technology. To inform security experts and policy decision makers on this matter, we review what is currently known on the potential uses and risks of quantum computers, leveraging current research literature. The maturity of currently-available quantum computers is not yet at a level such that they can be used in production for large-scale, industrially-relevant problems, and they are not believed to currently pose security risks. We identify 2 large-scale trends -- new approximate methods (variational algorithms, error mitigation, and circuit knitting) and the commercial exploration of business-relevant quantum applications -- which, together, may enable useful and practical quantum computing in the near future. Crucially, these methods do not appear likely to change the required resources for cryptanalysis on currently-used cryptosystems. From an analysis we perform of the current and known algorithms for cryptanalysis, we find they require circuits of a size exceeding those that can be run by current and near-future quantum computers (and which will require error correction), though we acknowledge improvements in quantum algorithms for these problems are taking place in the literature. In addition, the risk to cybersecurity can be well-managed by the migration to new, quantum-safe cryptographic protocols, which we survey and discuss. Given the above, we conclude there is a credible expectation that quantum computers will be capable of performing computations which are economically-impactful before they will be capable of performing ones which are cryptographically-relevant.
In this paper we study single qutrit circuits consisting of words over the Clifford$+D$ cyclotomic gate set, where $D=\text{diag}(\pm\xi^{a},\pm\xi^{b},\pm\xi^{c})$, $\xi$ is a primitive $9$-th root of unity and $a,b,c$ are integers. We characterize classes of qutrit unit vectors $z$ with entries in $\mathbb{Z}[\xi, \frac{1}{\chi}]$ based on the possibility of reducing their smallest denominator exponent (sde) with respect to $\chi := 1 - \xi,$ by acting an appropriate gate in Clifford$+D$. We do this by studying the notion of `derivatives mod $3$' of an arbitrary element of $\mathbb{Z}[\xi]$ and using it to study the smallest denominator exponent of $HDz$ where $H$ is the qutrit Hadamard gate and $D$. In addition, we reduce the problem of finding all unit vectors of a given sde to that of finding integral solutions of a positive definite quadratic form along with some additional constraints. As a consequence we prove that the Clifford$+D$ gates naturally arise as gates with sde $0$ and $3$ in the group $U(3,\mathbb{Z}[\xi, \frac{1}{\chi}])$ of $3 \times 3$ unitaries with entries in $\mathbb{Z}[\xi, \frac{1}{\chi}]$. We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford$+R$ and recover the previous exact synthesis algorithm in \cite{kmm}. The framework developed to formulate qutrit gate synthesis for Clifford$+D$ extends to qudits of arbitrary prime power.
We describe a quantum algorithm for finding the smallest eigenvalue of a Hermitian matrix. This algorithm combines Quantum Phase Estimation and Quantum Amplitude Estimation to achieve a quadratic speedup with respect to the best classical algorithm in terms of matrix dimensionality, i.e., $\widetilde{\mathcal{O}}(\sqrt{N}/\epsilon)$ black-box queries to an oracle encoding the matrix, where $N$ is the matrix dimension and $\epsilon$ is the desired precision. In contrast, the best classical algorithm for the same task requires $\Omega(N)\text{polylog}(1/\epsilon)$ queries. In addition, this algorithm allows the user to select any constant success probability. We also provide a similar algorithm with the same runtime that allows us to prepare a quantum state lying mostly in the matrix's low-energy subspace. We implement simulations of both algorithms and demonstrate their application to problems in quantum chemistry and materials science.
In this work, we present a generic approach to transform CSS codes by building upon their equivalence to phase-free ZX diagrams. Using the ZX calculus, we demonstrate diagrammatic transformations between encoding maps associated with different codes. As a motivating example, we give explicit transformations between the Steane code and the quantum Reed-Muller code, since by switching between these two codes, one can obtain a fault-tolerant universal gate set. To this end, we propose a bidirectional rewrite rule to find a (not necessarily transversal) physical implementation for any logical ZX diagram in any CSS code. Then we focus on two code transformation techniques: code morphing, a procedure that transforms a code while retaining its fault-tolerant gates, and gauge fixing, where complimentary codes can be obtained from a common subsystem code (e.g., the Steane and the quantum Reed-Muller codes from the [[15,1,3,3]] code). We provide explicit graphical derivations for these techniques and show how ZX and graphical encoder maps relate several equivalent perspectives on these code-transforming operations.
While implementing a quantum algorithm it is crucial to reduce the quantum resources, in order to obtain the desired computational advantage. For most fault-tolerant quantum error-correcting codes the cost of implementing the non-Clifford gate is the highest among all the gates in a universal fault-tolerant gate set. In this paper we design provable algorithm to determine T-count of any $n$-qubit ($n\geq 1$) unitary $W$ of size $2^n\times 2^n$, over the Clifford+T gate set. The space and time complexity of our algorithm are $O\left(2^{2n}\right)$ and $O\left(2^{2n\mathcal{T}_{\epsilon}(W)+4n}\right)$ respectively. $\mathcal{T}_{\epsilon}(W)$ ($\epsilon$-T-count) is the (minimum possible) T-count of an exactly implementable unitary $U$ i.e. $\mathcal{T}(U)$, such that $d(U,W)\leq\epsilon$ and $\mathcal{T}(U)\leq\mathcal{T}(U')$ where $U'$ is any exactly implementable unitary with $d(U',W)\leq\epsilon$. $d(.,.)$ is the global phase invariant distance. Our algorithm can also be used to determine the (minimum possible) T-depth of any multi-qubit unitary and the complexity has exponential dependence on $n$ and $\epsilon$-T-depth. This is the first algorithm that gives T-count or T-depth of any multi-qubit ($n\geq 1$) unitary. For small enough $\epsilon$, we can synthesize the T-count and T-depth-optimal circuits. Our results can be used to determine the minimum count (or depth) of non-Clifford gates required to implement any multi-qubit unitary with a universal gate set consisting of Clifford and non-Clifford gates like Clifford+CS, Clifford+V, etc. To the best of our knowledge, there were no such optimal-synthesis algorithm for arbitrary multi-qubit unitaries in any universal gate set.
We investigate the problem of synthesizing T-depth optimal quantum circuits over the Clifford+T gate set. First we construct a special subset of T-depth 1 unitaries, such that it is possible to express the T-depth-optimal decomposition of any unitary as product of unitaries from this subset and a Clifford (up to global phase). The cardinality of this subset is at most $n\cdot 2^{5.6n}$. We use nested meet-in-the-middle (MITM) technique to develop algorithms for synthesizing provably \emph{depth-optimal} and \emph{T-depth-optimal} circuits for exactly implementable unitaries. Specifically, for synthesizing T-depth-optimal circuits, we get an algorithm with space and time complexity $O\left(\left(4^{n^2}\right)^{\lceil d/c\rceil}\right)$ and $O\left(\left(4^{n^2}\right)^{(c-1)\lceil d/c\rceil}\right)$ respectively, where $d$ is the minimum T-depth and $c\geq 2$ is a constant. This is much better than the complexity of the algorithm by Amy et al.(2013), the previous best with a complexity $O\left(\left(3^n\cdot 2^{kn^2}\right)^{\lceil \frac{d}{2}\rceil}\cdot 2^{kn^2}\right)$, where $k>2.5$ is a constant. We design an even more efficient algorithm for synthesizing T-depth-optimal circuits. The claimed efficiency and optimality depends on some conjectures, which have been inspired from the work of Mosca and Mukhopadhyay (2020). To the best of our knowledge, the conjectures are not related to the previous work. Our algorithm has space and time complexity $poly(n,2^{5.6n},d)$ (or $poly(n^{\log n},2^{5.6n},d)$ under some weaker assumptions).
While mapping a quantum circuit to the physical layer one has to consider the numerous constraints imposed by the underlying hardware architecture. Connectivity of the physical qubits is one such constraint that restricts two-qubit operations, such as CNOT, to "connected" qubits. SWAP gates can be used to place the logical qubits on admissible physical qubits, but they entail a significant increase in CNOT-count. In this paper we consider the problem of reducing the CNOT-count in Clifford+T circuits on connectivity constrained architectures, like noisy intermediate-scale quantum (NISQ) computing devices. We "slice" the circuit at the position of Hadamard gates and "build" the intermediate {CNOT,T} sub-circuits using Steiner trees, significantly improving on previous methods. We compared the performance of our algorithms while mapping different benchmark and random circuits to some well-known architectures such as 9-qubit square grid, 16-qubit square grid, Rigetti 16-qubit Aspen, 16-qubit IBM QX5 and 20-qubit IBM Tokyo. Our methods give less CNOT-count compared to Qiskit and TKET transpiler as well as using SWAP gates. Assuming most of the errors in a NISQ circuit implementation are due to CNOT errors, then our method would allow circuits with few times more CNOT gates be reliably implemented than the previous methods would permit.
Hyperledger Fabric is a prominent and flexible solution for building permissioned distributed ledger platforms. Access control and identity management relies on a Membership Service Provider (MSP) whose cryptographic interface only handles standard PKI methods for authentication: RSA and ECDSA classical signatures. Also, MSP-issued credentials may use only one signature scheme, tying the credential-related functions to classical single-signature primitives. RSA and ECDSA are vulnerable to quantum attacks, with an ongoing post-quantum standardization process to identify quantum-safe drop-in replacements. In this paper, we propose a redesign of Fabric's credential-management procedures and related specifications in order to incorporate hybrid digital signatures, protecting against both classical and quantum attacks using one classical and one quantum-safe signature. We create PQFabric, an implementation of Fabric with hybrid signatures that integrates with the Open Quantum Safe (OQS) library. Our implementation offers complete crypto-agility, with the ability to perform live migration to a hybrid quantum-safe blockchain and select any existing OQS signature algorithm for each node. We perform comparative benchmarks of PQFabric with each of the NIST candidates and alternates, revealing that long public keys and signatures lead to an increase in hashing time that is sometimes comparable to the time spent signing or verifying messages itself. This is a new and potentially significant issue in the migration of blockchains to post-quantum signatures.
This work focuses on reducing the physical cost of implementing quantum algorithms when using the state-of-the-art fault-tolerant quantum error correcting codes, in particular, those for which implementing the T gate consumes vastly more resources than the other gates in the gate set. More specifically, we consider the group of unitaries that can be exactly implemented by a quantum circuit consisting of the Clifford+T gate set, a universal gate set. Our primary interest is to compute a circuit for a given $n$-qubit unitary $U$, using the minimum possible number of T gates (called the T-count of unitary $U$). We consider the problem COUNT-T, the optimization version of which aims to find the T-count of $U$. In its decision version the goal is to decide if the T-count is at most some positive integer $m$. Given an oracle for COUNT-T, we can compute a T-count-optimal circuit in time polynomial in the T-count and dimension of $U$. We give a provable classical algorithm that solves COUNT-T (decision) in time $O\left(N^{2(c-1)\lceil\frac{m}{c}\rceil}\text{poly}(m,N)\right)$ and space $O\left(N^{2\lceil\frac{m}{c}\rceil}\text{poly}(m,N)\right)$, where $N=2^n$ and $c\geq 2$. This gives a space-time trade-off for solving this problem with variants of meet-in-the-middle techniques. We also introduce an asymptotically faster multiplication method that shaves a factor of $N^{0.7457}$ off of the overall complexity. Lastly, beyond our improvements to the rigorous algorithm, we give a heuristic algorithm that outputs a T-count-optimal circuit and has space and time complexity $\text{poly}(m,N)$, under some assumptions. While our heuristic method still scales exponentially with the number of qubits (though with a lower exponent, there is a large improvement by going from exponential to polynomial scaling with $m$.
There have been several efforts to apply quantum SAT solving methods to factor large integers. While these methods may provide insight into quantum SAT solving, to date they have not led to a convincing path to integer factorization that is competitive with the best known classical method, the Number Field Sieve. Many of the techniques tried involved directly encoding multiplication to SAT or an equivalent NP-hard problem and looking for satisfying assignments of the variables representing the prime factors. The main challenge in these cases is that, to compete with the Number Field Sieve, the quantum SAT solver would need to be superpolynomially faster than classical SAT solvers. In this paper the use of SAT solvers is restricted to a smaller task related to factoring: finding smooth numbers, which is an essential step of the Number Field Sieve. We present a SAT circuit that can be given to quantum SAT solvers such as annealers in order to perform this step of factoring. If quantum SAT solvers achieve any speedup over classical brute-force search, then our factoring algorithm is faster than the classical NFS.
Measuring the expectation value of Pauli operators on prepared quantum states is a fundamental task in a multitude of quantum algorithms. Simultaneously measuring sets of operators allows for fewer measurements and an overall speedup of the measurement process. We investigate the task of partitioning a random subset of Pauli operators into simultaneously-measurable parts. Using heuristics from coloring random graphs, we give an upper bound for the expected number of parts in our partition. We go on to conjecture that allowing arbitrary Clifford operators before measurement, rather than single-qubit operations, leads to a decrease in the number of parts which is linear with respect to the lengths of the operators. We give evidence to confirm this conjecture and comment on the importance of this result for a specific near-term application: speeding up the measurement process of the variational quantum eigensolver.
Currently available quantum computing hardware platforms have limited 2-qubit connectivity among their addressable qubits. In order to run a generic quantum algorithm on such a platform, one has to transform the initial logical quantum circuit describing the algorithm into an equivalent one that obeys the connectivity restrictions. In this work we construct a circuit synthesis scheme that takes as input the qubit connectivity graph and a quantum circuit over the gate set generated by $\{\text{CNOT},R_{Z}\}$ and outputs a circuit that respects the connectivity of the device. As a concrete application, we apply our techniques to Google's Bristlecone 72-qubit quantum chip connectivity, IBM's Tokyo 20-qubit quantum chip connectivity, and Rigetti's Acorn 19-qubit quantum chip connectivity. In addition, we also compare the performance of our scheme as a function of sparseness of randomly generated quantum circuits. Note: Recently, the authors of arXiv:1904.00633 independently presented a similar optimization scheme. Our work is independent of arXiv:1904.00633, being a longer version of the seminar presented by Beatrice Nash at the Dagstuhl Seminar 18381: Quantum Programming Languages, pg. 120, September 2018, Dagstuhl, Germany, slide deck available online at https://materials.dagstuhl.de/files/18/18381/18381.BeatriceNash.Slides.pdf.
Heat-bath algorithmic cooling (HBAC) provides algorithmic ways to improve the purity of quantum states. These techniques are complex iterative processes that change from each iteration to the next and this poses a significant challenge to implementing these algorithms. Here, we introduce a new technique that on a fundamental level, shows that it is possible to do algorithmic cooling and even reach the cooling limit without any knowledge of the state and using only a single fixed operation, and on a practical level, presents a more feasible and robust alternative for implementing HBAC. We also show that our new technique converges to the asymptotic state of HBAC and that the cooling algorithm can be efficiently implemented; however, the saturation could require exponentially many iterations and remains impractical. This brings HBAC to the realm of feasibility and makes it a viable option for realistic application in quantum technologies.
Quantum algorithms can break factoring and discrete logarithm based cryptography and weaken symmetric cryptography and hash functions. In order to estimate the real-world impact of these attacks, apart from tracking the development of fault-tolerant quantum computers it is important to have an estimate of the resources needed to implement these quantum attacks. For attacking symmetric cryptography and hash functions, generic quantum attacks are substantially less powerful than they are for today's public-key cryptography. So security will degrade gradually as quantum computing resources increase. At present, there is a substantial resource overhead due to the cost of fault-tolerant quantum error correction. We provide estimates of this overhead using state-of-the-art methods in quantum fault-tolerance. We use state-of-the-art optimized circuits, though further improvements in their implementation would also reduce the resources needed to implement these attacks. To bound the potential impact of further circuit optimizations we provide cost estimates assuming trivial-cost implementations of these functions. These figures indicate the effective bit-strength of the various symmetric schemes and hash functions based on what we know today (and with various assumptions on the quantum hardware), and frame the various potential improvements that should continue to be tracked. As an example, we also look at the implications for Bitcoin's proof-of-work system. For many of the currently used asymmetric (public-key) cryptographic schemes based on RSA and elliptic curve discrete logarithms, we again provide cost estimates based on the latest advances in cryptanalysis, circuit compilation and quantum fault-tolerance theory. These allow, for example, a direct comparison of the quantum vulnerability of RSA and elliptic curve cryptography for a fixed classical bit strength.