Let’s imagine that we are given a quantum computer with 100 logical qubits, and let’s also assume that we have high gate fidelity (i.e. applying a gate won’t introduce major errors in our computation). This means that we can run all the algorithm that we want. An idyllic situation like this won’t probably happen in the near future (let’s say 5 years). Even if now we have the first prototypes of quantum computers with the first dozen of quibts, those qubits are not stable, and therefore the computation we can do is pretty limited: in fact, these prototypes aren’t able to perform error-free computations (there’s no error correction yet), and that the computation won’t be “long” as we want: we will be able to apply a limited number of gate before the system decohere.

The question is the following: can we compete in solving linear system of equations with a classical computer? Can we use it to run HHL algorithm? Let’s recall that for HHL we need 1 qubit register for the ancilla, a register for the output of phase estimation in the Hamiltonian simulation (that will store the superposition of the eigenvalues) and the rest of the qubit can store the input register. We assume to have logical qubits in our comparison.

We’ll see what happen when we change precision for floating point operations: 32, 64bit. The sparsity of the matrix is supposed to be small. Since we want to be as close as possible as real cases, let’s take a famous example of matrix considered sparse. The product-user matrix that websites like Amazon, or Netflix use to run Recommendation Algorithms: rows represent the users of the service, while columns are the products. The rows are empty except where an user purchased a specific product or watched a particular movie. Let’s say that an educated guess for the sparsity of the matrix is $100$.

The upper horizontal line is an estimate of the space in TB of the hard-disks for the whole Google (Cirrusinsight, n.d.) (13 EB), while the lower one is an estimate for the storage need to store the images of Google Maps (Mesarina, n.d.) (43 PB).

Let’s do an example to show what the software is plotting for $100$ qubits. In HHL we need an ancilla qubit. So we have 99 qubits. To get $64$ bit precision, we need to allocate 64 qubits: this is the phase estimation of the Hamiltonian simulation step. So now we are left with just $35$ qubits. With $35$ qubits we can span a Hilbert space of dimension $2^{35}$: this allow us to encode a vector of data with the same number of components. Suppose our vector of known values has $64$ bits floating points: classically, the cost for storing this amount of data we need $2^{35}*64$ bits, which are $0.549.5$TB (Terabytes). Summing up the cost for storing a $2^{35} \times 2^{35}$ matrix with sparsity $100$, we get $27$ TB.

Remember that each component of the vector will be encoded as probability amplitude in our quantum register. This imply that our precision in modifying a qubit need to grow, along with the number fo qubit and the fidelity of the gates. Here we just focused on the computational capabilities of a small quantum computer with respect to HHL algorithm. Don’t forget that for HHL we will need to store the matrix to invert as in the classical case (in form of QRAM or other oracle). [Here](https://github.com/Scinawa/space_estimation_hhl) the code for generating the plot.

Cirrusinsight. n.d. “How Much Data Does Google Store?”

Mesarina, Malena. n.d. “How Much Storage Space Do You Need for Google
Maps?”