Is the “Numerical Data” Here Classical Data?

The short answer is: yes, in the parent document the phrase “numerical data” mostly means classical numerical data, even when those numbers are encoded into a quantum device. The numbers are part of the problem instance: matrix entries, payoff values, probabilities, angles, samples, model parameters, and so on. They are not themselves mysterious quantum states. They are ordinary quantitative values that the algorithm must somehow access.

But there is an important subtlety. A number can be classical as information while being stored or accessed through a quantum representation. That distinction is central to the parent document.

If the problem contains a real number such as

\[ x = 0.734918\ldots, \]

then \(x\) is a classical value: it is a parameter one could, in principle, write down with digits, approximate by a bit string, store in ordinary memory, or use to calibrate a physical operation. But a quantum algorithm might receive this same classical value in several different physical forms. It might be stored digitally in qubits as a binary string. It might appear as a phase angle in a unitary rotation. It might be hidden inside an oracle, a state-preparation circuit, or a block-encoding. The parent document’s concern is that these representations do not all make the same amount of precision equally available.

Classical information versus quantum representation

A useful way to separate the ideas is this:

The data value may be classical, while the carrier of that value may be quantum.

For example, suppose a number \(x \in [0,1)\) is approximated by an \(n\)-bit binary string,

\[ x \approx 0.b_1 b_2 \cdots b_n. \]

The bits \(b_1,\ldots,b_n\) are classical information. A quantum computer can store those same bits in computational-basis states such as

\[ |b_1 b_2 \cdots b_n\rangle. \]

Although the storage medium is quantum, the stored content is still classical in the usual sense: it is a definite basis label. If the register is known to be one of the computational-basis states, then it is being used as reversible digital memory, not as an unknown quantum superposition. This is standard in quantum computation: qubits can carry classical bit strings as a special case of quantum states [Nielsen and Chuang 2010].

Now compare that with phase encoding. Instead of storing the bits of \(x\), one might define a unitary

\[ U_x = e^{-i x G}, \]

where \(G\) is a Hermitian generator. Here \(x\) is still a classical parameter. It is a number chosen by the problem instance. But the algorithm does not necessarily possess the binary expansion of \(x\). It may only be allowed to call the physical transformation \(U_x\). In that case, the information is classical in origin but operationally available only through quantum interaction with a parameter-dependent channel.

That is why the parent document says the numbers “must be made physically available to the algorithm in some representation.” The document is not saying that numerical data automatically become quantum data. It is saying that even classical numbers need a physical interface, and the cost of that interface matters.

When numerical data are classical

Most of the parent document’s examples are naturally classical numerical data. Matrix entries, payoff values, correlations, fixed-point bit strings, phase angles, thresholds, and model parameters are normally values describing a problem instance. In a financial risk calculation, for example, asset correlations and loss thresholds are classical model inputs. In a linear-algebra problem, the entries of a matrix \(A\) are classical numbers unless the problem explicitly says that \(A\) is available only through an unknown quantum process.

Quantum algorithms often assume classical data are accessible through idealized oracles. For example, an oracle might implement

\[ O_f |j\rangle |0\rangle = |j\rangle |f(j)\rangle, \]

where \(j\) is an index and \(f(j)\) is a classical value stored reversibly in a target register. The oracle is a quantum operation, but the function \(f\) is classical data. Similarly, quantum algorithms for matrix problems often assume access to a block-encoding of a matrix; the matrix entries are classical problem data, while the block-encoding is a unitary representation through which the algorithm interacts with that data [Gilyén et al. 2019].

This is the sense in which the highlighted passage should be read. “Numerical data” means the quantitative content of the problem. In the usual setting, that content is classical.

When numerical data may instead be quantum data

There is another possible meaning, but it is not the dominant one in the parent document. Sometimes the input to a quantum algorithm is not a classical description of numbers but an actual unknown quantum state,

\[ |\psi\rangle = \sum_j \alpha_j |j\rangle. \]

The amplitudes \(\alpha_j\) are complex numbers in the mathematical description of the state. But if the algorithm is given only copies of an unknown state \(|\psi\rangle\), then those amplitudes are not classical data available for free. They are quantum-state parameters. They cannot be directly read out without measurement, and measurement generally gives only samples, not full amplitude values. Moreover, an unknown quantum state cannot be copied arbitrarily because of the no-cloning theorem [Wootters and Zurek 1982; Nielsen and Chuang 2010].

So the word “amplitudes” needs care. If a paper says, “prepare a state whose amplitudes encode a known classical vector,” then the vector entries are classical numerical data encoded into a quantum state. But if the input is an unknown physical state and the task is to estimate its amplitudes, then the amplitudes are part of quantum data. The parent document appears mainly concerned with the first situation: known or specified numerical values that a quantum algorithm is assumed to access.

A compact distinction is:

The parent document’s warning mainly applies to the first three cases.

Why this distinction matters for the parent document

The parent document is not merely asking whether numbers are “classical” or “quantum” in a philosophical sense. It is asking whether the useful precision of those numbers has been honestly counted.

Suppose a 64-bit classical value \(x\) is stored digitally. Then, ignoring fault-tolerance and implementation overheads, the representation explicitly contains 64 bits of information. But suppose instead that \(x\) is encoded as a phase,

\[ R_z(2\pi x). \]

Mathematically, the angle \(2\pi x\) may contain arbitrarily many digits. Physically, however, an algorithm given access to this rotation cannot simply read all those digits. To distinguish two nearby values \(x\) and \(x+\Delta x\), it must perform operations whose statistics differ enough to be observed. In ideal phase-estimation and metrology settings, resolving smaller \(\Delta x\) generally requires more coherent uses of the parameter-dependent operation; this is the broad content of quantum estimation theory and Heisenberg-limited scaling [Helstrom 1976; Giovannetti, Lloyd, and Maccone 2006].

So if the user asks, “Is this numerical data classical data?”, the most accurate answer is:

The numerical data are usually classical problem-instance values, but the parent document is studying what happens when those classical values are supplied through quantum encodings.

That is exactly where the cost question appears. A classical number encoded as a quantum phase may look compact, but the missing digital bits may reappear as query complexity, calibration cost, or estimation cost.

The safest interpretation

The safest reading of the highlighted passage is therefore not “numerical data means quantum data.” It is closer to this:

“Numerical data” means classical quantitative information that the quantum algorithm needs in order to solve the instance, regardless of whether that information is stored in ordinary bits, qubit basis states, quantum phases, or more complicated oracle and block-encoding interfaces.

There are exceptions if the parent document later discusses unknown quantum states as inputs. In that case, some numerical quantities, such as amplitudes or density-matrix parameters, would be quantum data in the operational sense. But from the supplied text, the main target is classical numerical information made available to a quantum algorithm.

That distinction helps preserve the parent document’s main point: the hard issue is not whether the numbers are “classical” in origin, but whether their required precision is physically and computationally available to the algorithm.

References

Gilyén, András, Yuan Su, Guang Hao Low, and Nathan Wiebe. 2019. “Quantum Singular Value Transformation and Beyond: Exponential Improvements for Quantum Matrix Arithmetics.” In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, 193–204.

Giovannetti, Vittorio, Seth Lloyd, and Lorenzo Maccone. 2006. “Quantum Metrology.” Physical Review Letters 96: 010401. DOI: 10.1103/PhysRevLett.96.010401.

Helstrom, Carl W. 1976. Quantum Detection and Estimation Theory. Academic Press.

Nielsen, Michael A., and Isaac L. Chuang. 2010. Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press.

Wootters, William K., and Wojciech H. Zurek. 1982. “A Single Quantum Cannot Be Cloned.” Nature 299: 802–803. DOI: 10.1038/299802a0.