Quantum persistent homology

Abstract

Persistent homology is a powerful mathematical tool that summarizes useful information about the shape of data allowing one to detect persistent topological features while one adjusts the resolution. However, the computation of such topological features is often a rather formidable task necessitating the sub-sampling the underlying data. To remedy this, we develop an efficient quantum computation of persistent Betti numbers, which track topological features of data across different scales. Our approach employs a persistent Dirac operator whose spectrum relates to that of the persistent combinatorial Laplacian, and thus allows us to recover the persistent Betti numbers which capture the persistent features of data. In addition, our algorithm can also extract the non-harmonic spectra of the Laplacian, which can be used for data analysis as well. We also test our algorithm on a point cloud data.

Appendices

Appendix A: Membership oracle

We can encode the order k of a simplex $\sigma$ in a state $|k\rangle$ ($k=0,1,\dots ,n-1$) by starting from $|0\rangle$ and performing permutations $0\rightarrow 1 \rightarrow \dots \rightarrow n-1 \rightarrow 0$, conditional upon the corresponding digit of $\sigma$ being 1. Thus, we perform k permutations mapping $|0\rangle \rightarrow |k\rangle$. This can be implemented efficiently because the permutation is a 1-sparse matrix.

To encode the scale $\epsilon$ we need information on the data points that can be stored in quantum parallel in QRAM, if it is available, and accessed efficiently (Giovannetti et al. 2008a, b). For any $i,j = 1,2,\dots , n$, $\text{ QRAM }|i\rangle |j\rangle |0\rangle =|i\rangle |j\rangle |d(i,j)\rangle$, where d(i, j) is the distance between points i and j. Notice that the size of the memory is only logarithmic on the number of data points. We introduce a register of qubits to record the parameter $\epsilon$ as $|\epsilon \rangle$. We need to know when $d(i,j) \le \epsilon$ to form a VR complex. This information will be stored in a qubit initially in the state $|0\rangle$, and flipped if the membership condition is satisfied. This is implemented with a unitary test that uses the qubit registers storing d(i, j) and $\epsilon$ as controls to flip the last qubit,

$$\begin{aligned} U_{\text {test}}^\epsilon |d(i,j)\rangle |\epsilon \rangle |0\rangle = |d(i,j)\rangle |\epsilon \rangle |a^\epsilon (i,j)\rangle \, \ \ a^\epsilon (i,j) = \left\{ \begin{array}{ll} 0, &{}\quad d(i,j) > \epsilon \\ 1, &{} \quad d(i,j) \le \epsilon \end{array} \right. \end{aligned}$$

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Next, in order to know if $\sigma \in S^{\epsilon }$, we must check if $d(i,j) \le \epsilon$ for all (i, j) pairs such that $v_i = v_j = 1$. To this end, we make ${\mathcal {O}} (k^2)$ calls to QRAM, where k is the dimension of $\sigma$. For each pair (i, j), we use $|\sigma \rangle$ as control to call QRAM and apply the test provided $v_i = v_j = 1$, $\text {QRAM}^\dagger U_{\text {test}}^\epsilon \text {QRAM} |\sigma \rangle |i\rangle |j\rangle |0\rangle |\epsilon \rangle |0\rangle = |\sigma \rangle |i\rangle |j\rangle |0\rangle |\epsilon \rangle |a^\epsilon (i,j)\rangle .$ The membership of $\sigma$ in the VR complex, $S^\epsilon$, is decided if for all (i, j) we end up with $a^\epsilon (i,j) =1$.

Appendix B: Grover’s algorithm

Here we review the salient features of amplitude amplification (Brassard et al. 2002) and Grover’s search algorithm (Grover 1998) which are needed for the implementation of the projection $P^\epsilon$ (Eq. (12)), for completeness.

Let $|\Psi _k\rangle \in \mathcal {H}_{k}$ be a state in the span of the k-simplex states. We wish to construct the normalized projected state $|\Psi _k^\epsilon \rangle = \frac{P^\epsilon |\Psi _k\rangle }{\left\Vert P^\epsilon |\Psi _k\rangle \right\Vert }\in \mathcal {H}_{k}^{\epsilon }$, assuming that it exists. To this end, we introduce the unitary operator $U_G = -U_{\Psi _k} U^\epsilon$, with $U_{\Psi _k} = I - 2 |\Psi _k\rangle \langle \Psi _k|$ and $U^\epsilon = I - 2P^\epsilon$. Since $\mathcal {H}_{k}^{\epsilon }$ is a closed subspace, we may write $|\Psi _k\rangle$ as $|\Psi _k\rangle = \sin \theta |\Psi _k^\epsilon \rangle + \cos \theta |\bar{\Psi }_k^\epsilon \rangle$, where $|\Psi _{k}^{\epsilon }\rangle \in \mathcal {H}_{k}^{\epsilon }$ and $|\bar{\Psi }_{k}^{\epsilon }\rangle \in \mathcal {H}_{k}^{\epsilon \perp }$. Notice that $\sin \theta = \left\Vert P^\epsilon |\Psi _k\rangle \right\Vert$. We can think of $|\Psi _k\rangle$ as the vector $( \sin \theta , \cos \theta )^{T}$ in the two-dimensional space spanned by $\{ |\Psi _k^\epsilon \rangle , |\bar{\Psi }_k^\epsilon \rangle \}$, then $U_G$ acts as a rotation by an angle $2\theta$. Applying it K times, we obtain the state $U_G^K |\Psi _k\rangle = \sin (2K+1)\theta |\Psi _k^\epsilon \rangle + \cos (2K+1)\theta |\bar{\Psi }_k^\epsilon \rangle$. This is close to the desired state for $(2K+1)\theta \approx \frac{\pi }{2}$. Therefore, the number of Grover steps needed is $K = \lfloor \frac{\pi }{4\theta } \rfloor$. As discussed in Schmidhuber and Lloyd (2022), K could be exponential on the number of data points if $\left\Vert P^{\epsilon } |\Psi _{k}\rangle \right\Vert$ is small, for example when the number of simplices present in $S_{k}^{\epsilon }$ is only polynomial on the number data points.

Appendix C: Implementation of an exponential operator

Relying on Rebentrost et al. (2018), we review the construction of the exponential operator $e^{it B_k^{\epsilon ,\epsilon '}[\xi ]}$, where the shifted Dirac matrix $B_k^{\epsilon ,\epsilon '}[\xi ]$ is defined in Eq. (17). This construction is needed for the phase estimation algorithm described in Sect. 3.4 (See Eq. (20)). We start by constructing the $\text {SWAP}_{{B}}$ operator from the shifted Dirac operator $B_k^{\epsilon ,\epsilon '}[\xi ]$,

$$\begin{aligned} {\mathcal {S}} \equiv \text {SWAP}_{{B}}{} & {} = \sum _{\psi ,\phi } B_k^{\epsilon ,\epsilon '}[\xi ](\psi ,\phi ) |\phi \rangle \langle \psi | \otimes |\psi \rangle \langle \phi | \, \ \ B_k^{\epsilon ,\epsilon '}[\xi ](\psi ,\phi ) \nonumber \\{} & {} \quad = \langle \psi | B_k^{\epsilon ,\epsilon '}[\xi ] |\phi \rangle \, \end{aligned}$$

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where $|\psi \rangle = |0\rangle |\psi _{0}\rangle + |1\rangle |\psi _{1}\rangle + |2\rangle |\psi _{2}\rangle$, with $|\psi _0\rangle \in {\mathcal {H}}_{k-1}$, $|\psi _1\rangle \in {\mathcal {H}}_{k}$, $|\psi _2\rangle \in {\mathcal {H}}_{k+1}$, and similarly for $|\phi \rangle$. Let $\varvec{N}$ be the dimensionality of the Hilbert space in which $|\psi \rangle$ and $|\phi \rangle$ live and $\{ |e_a\rangle , a = 1,\dots ,\varvec{N} \}$ an orthonormal basis for the Hilbert space under consideration. With the choice $\xi = 1$, all matrix elements of the $\varvec{N}\times \varvec{N}$ matrix $B_k^{\epsilon ,\epsilon '}[\xi ](e_a,e_b)\in \{ 0, \pm 1 \}$ and the matrix ${\mathcal {S}}$ can be efficiently constructed. Then we construct the exponential $\text {SWAP}_{{B}}$ operator $e^{i\Delta t {\mathcal {S}}}$, which can be done efficiently because ${\mathcal {S}}$ is a one-sparse matrix. Next, we act on the state $|\varvec{s}\rangle \otimes |\Psi \rangle$, where $|\varvec{s}\rangle$ is the uniform state

$$\begin{aligned} |\varvec{s}\rangle = \frac{1}{\sqrt{\varvec{N}}} \sum _{a=1}^{\varvec{N}} |e_a\rangle , \end{aligned}$$

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and $|\Psi \rangle$ is an arbitrary state in the subspace on which $B_k^{\epsilon ,{{\epsilon }^{\prime }}}[\xi ]$ acts. After tracing over the space in which $|\varvec{s}\rangle$ lives, we obtain $\text {tr}_1 \left[ e^{-i \Delta t {\mathcal {S}}} |\varvec{s}\rangle \langle \varvec{s}| \otimes |\Psi \rangle \langle \Psi | e^{i\Delta t{\mathcal {S}}} \right] = e^{-i B_k^{\epsilon ,{{\epsilon }^{\prime }}}[\xi ] \Delta t/\varvec{N}} |\Psi \rangle \langle \Psi | e^{i B_k^{\epsilon ,{{\epsilon }^{\prime }}}[\xi ] \Delta t/\varvec{N}} + {\mathcal {O}} (\Delta t^2),$ which projects onto the state $e^{-i B_k^{\epsilon ,{{\epsilon }^{\prime }}}[\xi ] \Delta t/\varvec{N}} |\Psi \rangle$ up to second order in $\Delta t$. The desired state $e^{-it B_k^{\epsilon ,{{\epsilon }^{\prime }}}[\xi ]} |\Psi \rangle$ for finite t can be obtained by repeating the above construction as many times as needed.