Shor's algorithm is a canonical quantum supremacy target whose core operation relies on the Quantum Fourier Transform (QFT). In this note, we resolve an open question of Fang, Fenner, Green, Homer and Zhang from 2006 by showing that approximating QFT in constant depth, for any $n$-qubit modulus, necessarily requires the $n$-qubit Fanout operation.
Formally, let $\mathsf{QFT}_q$ be the gate acting on $n = \lceil \log q \rceil$ qubits that computes the QFT under modulus $q$. It is known that any $n$-qubit $\mathsf{QFT}_q$ can be implemented in constant depth using $\mathsf{FANOUT}_n$, i.e. $\mathsf{QFT}_q \in \mathsf{QAC}^0_f$. We prove the converse by using a $\mathsf{QFT}_q$ gate to construct a state of "non-negligible felinity". Consequently, $\mathsf{QFT}_q \in \mathsf{QAC}^0 \iff \mathsf{FANOUT}_n \in \mathsf{QAC^0}$.
In the case of $q = 2^n$, such as in Shor's, we approximate $\mathsf{FANOUT}_n$ using a single $\mathsf{QFT}_{2^n}$ gate and $O(1)$ two-qubit local gates, thus tying the feasibility of realizing Shor's algorithm with NISQ circuits to that of Fanout.
Shor's algorithm is a canonical quantum supremacy target whose core operation relies on the Quantum Fourier Transform (QFT). In this note, we resolve an open question of Fang, Fenner, Green, Homer and Zhang from 2006 by showing that approximating QFT in constant depth, for any n-qubit modulus, necessarily requires the n-qubit Fanout operation.
Formally, let QFTq be the gate acting on n=⌈logq⌉ qubits that computes the QFT under modulus q. It is known that any n-qubit QFTq can be implemented in constant depth using FANOUTn, i.e. QFTq∈QACf0. We prove the converse by using a QFTq gate to construct a state of "non-negligible felinity". Consequently, QFTq∈QAC0⟺FANOUTn∈QAC0.
In the case of q=2n, such as in Shor's, we approximate FANOUTn using a single QFT2n gate and O(1) two-qubit local gates, thus tying the feasibility of realizing Shor's algorithm with NISQ circuits to that of Fanout.
In 2005, Høyer and Špalek showed that constant-depth quantum circuits augmented with multi-qubit Fanout gates are quite powerful, able to compute a wide variety of Boolean functions as well as the quantum Fourier transform. They also asked what other multi-qubit gates could rival Fanout in terms of computational power, and suggested that the quantum Threshold gate might be one such candidate. Threshold is the gate that indicates if the Hamming weight of a classical basis state input is greater than some target value. We prove that Threshold is indeed powerful - there are polynomial-size constant-depth quantum circuits with Threshold gates that compute Fanout to high fidelity. Our proof is a generalization of a proof by Rosenthal that exponential-size constant-depth circuits with generalized Toffoli gates can compute Fanout. Our construction reveals that other quantum gates able to "weakly approximate" Parity can also be used as substitutes for Fanout.
In 2005, Høyer and Špalek showed that constant-depth quantum circuits augmented with multi-qubit Fanout gates are quite powerful, able to compute a wide variety of Boolean functions as well as the quantum Fourier transform. They also asked what other multi-qubit gates could rival Fanout in terms of computational power, and suggested that the quantum Threshold gate might be one such candidate. Threshold is the gate that indicates if the Hamming weight of a classical basis state input is greater than some target value. We prove that Threshold is indeed powerful - there are polynomial-size constant-depth quantum circuits with Threshold gates that compute Fanout to high fidelity. Our proof is a generalization of a proof by Rosenthal that exponential-size constant-depth circuits with generalized Toffoli gates can compute Fanout. Our construction reveals that other quantum gates able to "weakly approximate" Parity can also be used as substitutes for Fanout.
Shor's algorithm proved a significant milestone in quantum computing since it promises an exponential speedup over conventional algorithms for integer factorization, a problem critical to modern cryptography systems. This work covers implementation steps of Shor’s algorithm and analyzes Quantum Fourier Transform (QFT) usage for extracting periodicity from quantum superpositions. Firstly, theoretical foundations of algorithm are illustrated concentrating on QFT construction and operation within the setting of the quantum circuit. Thereafter, each implementation step such as qubit optimization, modular exponentiation, and circuit design is discussed. Our findings validate the theoretical effectiveness of the QFT in solving the period-finding subroutine, while also highlighting the practical difficulties and scaling constraints presented by existing quantum hardware. The paper ends with a summary of possible advancements and future paths for error mitigation strategies and quantum algorithm design.
The manuscript "A Comparison of Quantum and Traditional Fourier Transform Computations" discusses a very important and often overlooked aspect of quantum computing, namely a fair and detailed comparison of a quantum algorithm, its classical simulation, and its classical counterpart taking into account the complexity of I/O. Such an article is valuable and worth publishing. The current manuscript, however, still contains some inaccuracies that should be fixed and I will make suggestion on how the presentation can be improved.Let me first summarize the main result in my own words: a quantum Fourier transform (QFT) can calculate the Fourier transform of a vector with time complexity O(log2N), compared to the complexity O(NlogN) of a classical FFT. However, if one needs to read out the full vector instead, the complexity becomes O(N polylog(N)) again, without an advantage over the classical algorithm. It is actually worse than the author discusses, if one also takes into account the complexity of loading the initial state. Loading an initial classical data vector has complexity O(N), and this has to be repeated at every repetition, giving a complexity of O(N2polylogN), worse than classical. As the author correctly mentions, the QFT is thus a useful algorithm if the input data is prepared algorithmically, and limited sampling of the result vector is sufficient, as is the case for Shor's algorithm.This is a valuable observation that deserves a paper in CiSE, since these important considerations are not all known to people outside the quantum computing community, and are sometimes ignored even by quantum computing specialists. I thus want to encourage the author to improve the presentation to make it more accessible to a broader audience and fix a couple of technical flaws.Before discussing presentation issues, I want to address one technical flaw and a few points where more clarity is needed: This statement towards the end is too simplistic and needs to be clarified: "... if we want to measure each coefficient, we must redo our operations for each coefficient (since our wave function will collapse for every measurement)." The manuscript does not explain how to read out a specific coefficient. If one samples the wave function, one measures the result and gets a certain value (s1,⋯,sN) with a probability depending on the wave function. More complex amplitude-estimation algorithms are needed to read out specific coefficients, which then takes time O(N/ϵ). This needs to be better explained. This also raises another issue of normalization and precision. The quantum wave function has to be normalized to have an L2 norm of 1. We thus need to measure to a precision of epsilon divided by the L2 norm of the classical data, and if that increases with N the scaling is even worse. For example, if all entries are of order unity, the L2 norm is N. The one technical flaw in the paper is in preparing the input state to the QFT, and the complexity of the classical simulation of the quantum algorithm. There, the numbers are wrong. The classical complexity can easily be estimated by realizing that any sparse quantum gate (such as a 1-qubit or 1-qubit gate) can be simulated in time O(N)=O(2n), where N=2n. Thus, the overhead is just O(N), not O(N2), and the simulation complexity is just O(N) times the complexity of the quantum algorithm. This will be more apparent if, as I propose below, the author shows the quantum algorithm also through a sequence of quantum gates. However, note that in the simulation we then have the full vector, and thus do not incur the overhead of quantum state tomography. We need to simulate the algorithm only once and not O(N/ϵ) times. This means the simulation remains at O(Nlog2N) even if we read out all entries, which is not bad compared to the classical O(NlogN) of the FFT. The reason why the author seems to end with a different complexity is that his Matlab simulation does not simulate the quantum algorithm but the computation of the effect of the QFT applied to just one basis state. That is a suboptimal implementation. I thus propose to replace the Matlab code by a discussion of how a quantum gate (Hadamard or controlled phase rotation) can be implemented, and that will then nicely give the scaling discussed above.As mentioned in my summary at the beginning, the value of the paper can be increased if the author could also discuss the complexity of quantum state preparation from classical input data, e.g., following Shende-Bullock-Markov \citep{markov2006}, which has a complexity of O(Nlogϵ1). That means, that if the data is read from a classical vector (and not computed as, e.g., in Shor's algorithm), then the complexity of state preparation of O(Nlogϵ1) completely dominates the QFT itself, and if one furthermore wants to read out the full vector instead of sampling it, the complexity becomes O(N2log2(ϵ1)).In the conclusion the author writes that "QEC research is still in very early development and it is currently difficult to determine how the required resources for these corrections could scale with qubit usage." That is incorrect, as the overhead is pretty well known by now for certain QEC codes, such as the surface code. The asymptotic scaling of the overhead has long been known, and also detailed resource costs have been worked out for various algorithms. I suggest to focus the QEC section on the need for fault tolerance, and the large overhead associated with it. This is definitely not something that can be done on NISQ devices at an interesting scale. I would also use another reference for QEC than the Gil Kalai paper.Now, to suggestions for improved presentation:In the abstract, I suggest to present fewer technical details. "radix-2 DIT case of the Cooley-Tukey Algorithm" can just be called Fast Fourier transform, the QUBIT4MATLAB package does not need to be mentioned in the abstract, and neither does the "Master Theorem" have to be mentioned.In the abstract and in the rest of the paper, it is also better to talk about quantum speedup and not quantum supremacy. The paper is about asymptotic scaling (the author uses the big-O notation throughout), and not about supremacy, which is a specific size problem that is solved better on quantum hardware rather than classical hardware. A supremacy claim needs all constants to be worked out, and assumptions for the specific classical and quantum hardware. Replacing quantum supremacy by quantum speedup can fix this. The observations are still valuable and correct.The first section is called "The shortcomings of classical computing and the early developments of quantum computing". That title sounds strange to me. What are the shortcomings? The author may rather want to stress the additional capabilities of quantum computers. Given that most readers will not be physicists and have no experience with quantum mechanics, it would be valuable to expand this first section. Discuss equation (3) before equation (1), and then introduce equation (1), to show that this means that the state of a qubit is actually a two-dimensional complex vector, After that only go to the many-quit state. After equation 4 it may be good to talk about amplitudes first, and then probabilities. Saying that the state is determined by "the qubit simply has a probability...", ignores the all important phases that are the crucial difference between quantum computing and probabilistic classical computing. "Quantum gates, unlike classical gates, do not delete information and are fully reversible." Needs further explanation. This comes from the reversibility of the microscopic laws of physics. Here it may be worthwhile to mention that, as quantum gates are time evolution of a quantum system, they are unitary operators on the Hilbert space that was introduced above. The discussion in the next sentence about them not using energy is misleading as well. In fact, the control and operation of the quantum gate always needs energy. It is only the free evolution of a quantum system that conserves energy. The part on " subatomic systems such as a superposition of quantum states. " would also need more explanation. Does the author here want to describe specific implementations of qubits? If so, more detail is needed. If not, I propose to drop this part."The definition and use of the classical discrete Fourier transform": here is an abrupt jump from Shor's algorithm to the DFT. Since Shor's algorithm again gets mentioned later on as a case where the QFT is better than a classical DFT, it make sense to discuss more details of Shor's algorithm, and that in essence it solvers the factor problem by using a QFT to find the period of a function. In light of the above complexity discussion this is important: we don't deal with classical data that one has to load but with a state that is computed (the function f(x) = a^x mod N ), and one is not interested in the full vector but just in finding the period. This can be done just by measuring the result of the QFT, which in this particular case will return a random multiple of the base frequency, and with a few repeated measurements one can then extract the period).The Fourier transform should be well known to the reader, and it is thus not obvious to me that equation 6 is needed. I would instead introduce \omega in equation 5. For the discussion of the implementation I suggest to mention that the rest of the paper limits itself to the case of powers of 2, i.e. N=2^n, for which the radix-2 version applies. Since the implementation is well known for most computational scientists, it could be moved from to supplementary material, as can be the derivation of the O(N log N) complexity, which is also well known.In the discussion of the QFT, unfortunately the quantum algorithm is not presented. What is presented is the mathematical unitary operation. It would be very valuable for the reader to see an actual implementation of the quantum algorithm, i.e. the sequence of O(n^2) Hadamard gates and controlled rotations – either as a circuit drawing, or more useful as quantum code, either using a quantum programming language that allows loops (such as Q#), or as some quantum pseudo code with loops. Then the reader will see the sequence of O(n^2) operations and how it creates the unitary operation of the QFT."The implementation of the QFT in MATLAB...". This section can be improved as well., As discussed above, the Matlab code does not implement the quantum algorithm but rather implements the application of the unitary operation to one of the basis states. It would be more useful for the reader to instead get implementations of the Hadamard and controlled phase gates, which could then be combined with the above quantum algorithm (maybe all in Matlab even?) for a full simulation of the algorithm – and in time O(N n^2), not O(N^2 n^2)."Evaluating the complexity of the QFT Implementation" – I discussed the issues with the complexity estimates already above. If one replaces it by an estimate of the complexity of simulating the H and controlled phase gate each in O(N) operations then it both becomes simpler and the complexity becomes just O(N) times that of the quantum version"Evaluating the complexity of the theoretical form of the QFT Implementation”. I assume that by "theoretical" the author means the actual quantum operation? The comparison section needs to be fixed, since that's where there is the mentioned flaw based on the suboptimal Matlab code, and that will also change table 1. I suggest that table 1 could have three cases: A) ignoring state preparation and one only wants to sample the output a constant number of times; B) ignoring state preparation but one wants to read the full vector; C) including state preparation from classical data and one only wants to sample the output a constant number of times; D) including state preparation from classical data and one wants to read the full vectorThe reader will then see that only case A has a speedup, and that in the other cases even the simulation of the quantum algorithm is faster than the quantum algorithm on a quantum computer.The conclusions should then be updated, taking these slightly modified comparisons into account. The case of Shor's algorithm, which falls into category A) above, is indeed interesting but deserves a deeper explanation so that the reader understands that Shor's algorithm can compute the input with complexity O(N3) (up to log factors), and why only a few samples of the output are needed. That way, the exponential speedup over the FFT remains, and the super polynomial speedup over the name sieve. However, if classical data has to be loaded or should be returned then there is no speedup. I call this the I/O problem of quantum algorithms, which indeed—as the author writes—means that in these cases quantum computing is no better than classical computing.This is an important observation that the paper beautifully works out. Fixing the flaw in the simulation section and improving the presentation will make this a very nice paper.
The relevance of shallow-depth quantum circuits has recently increased, mainly due to their applicability to near-term devices. In this context, one of the main goals of quantum circuit complexity is to find problems that can be solved by quantum shallow circuits but require more computational resources classically. Our first contribution in this work is to prove new separations between classical and quantum constant-depth circuits. Firstly, we show a separation between constant-depth quantum circuits with quantum advice QNC0/qpoly , and AC0[p] , which is the class of classical constant-depth circuits with unbounded-fan in and MODp gates. Additionally, we show a separation between QAC0 , the circuit class containing Toffoli gates with unbounded control, and AC0[p] , when QAC0 is augmented with additional mid-circuit measurements and classical fanout. This establishes the first such separation for a shallow-depth quantum class that does not involve quantum fanout gates, while relying solely on finite quantum gate sets. Equivalently, this yields a separation between AC0[p] and [QNC0,AC0]2 , i.e., shallow quantum circuits interleaved with simple classical computation. Secondly, we consider QNC0 circuits with infinite-size gate sets. We show that these circuits, along with quantum prime modular gates or classical prime modular gates in combination with classical fanout, can implement threshold gates, showing that QNC0[p]=QTC0 . Finally, we also show that in the infinite-size gate set case, these quantum circuit classes for higher-dimensional Hilbert spaces do not offer any advantage to standard qubit implementations.
Classical circuits with unbounded fan-in can compute any Boolean function in constant depth when their size is unrestricted. We ask whether removing the restrictions on circuit size and ancillary qubits also allows quantum circuits built from arbitrary single-qubit gates and generalised Toffoli gates to implement every unitary in constant depth.
We give exact constant-depth constructions for arbitrary permutations of computational basis states, diagonal unitaries and the preparation of arbitrary pure states. These connect quantum state preparation to reversible classical computation and the preparation of probability distributions. With fanout in the gate set, they use exponentially many gates and ancillary qubits and return all ancillary qubits to zero. Replacing fanout by an exact circuit over the original gate set preserves constant depth, although the size bounds can become doubly exponential.
The implementation of arbitrary unitaries in constant depth remains open. We give equivalent formulations in terms of copying the vectors of a specified orthonormal basis, extracting their labels and implementing restricted families of unitaries. We also reduce arbitrary unitary implementation to that of traceless unitary involutions using one additional clean qubit. With adaptive measurements, gate teleportation gives depth proportional to the level of a gate in the Clifford hierarchy.
Towards arbitrary unitary implementation in constant depth, we use port-based teleportation: for input dimension d and M≥d2−1 ports, we construct a unitary circuit of depth O(d), independent of M and including resource preparation and port selection, with entanglement fidelity at least (1−(d2−1)/(2M))2. Thus, at fixed d, the approximation can be made arbitrarily accurate without increasing depth. Whether the dependence on d can also be removed remains open.