Quantum Wave Function Collapse for Procedural Content Generation

TL;DR

This work proposes a quantum version of the famous (classical) wave function collapse algorithm based on the idea that a quantum circuit can be prepared in such a way that it acts as a special-purpose random generator for content of a desired form.

Abstract

Quantum computers exhibit an inherent randomness, so it seems natural to consider them for procedural content generation. In this work, a quantum version of the famous (classical) wave function collapse algorithm is proposed. This quantum wave function collapse algorithm is based on the idea that a quantum circuit can be prepared in such a way that it acts as a special-purpose random generator for content of a desired form. The proposed method is presented theoretically and investigated experimentally on simulators and IBM Quantum devices.

Figures (11)

• Figure 1: Exemplary nearest neighbor adjacency configuration for on a two-dimensional grid consisting of $N=9$ segments. The adjacency relationship for each of the four directions right ($d=1$), up ($d=2$), left ($d=3$), and down ($d=4$) can be represented as a directed graph with adjacency matrix $\alpha^d$.
• Figure 2: Visualization of pattern-based rules. (a) Visualization scheme. The center tile corresponds to the target segment $i$ with a target value of $v$, whereas the adjacent tiles represent the required values $v_1,\dots,v_4 \in \{1,2\}$ of the four adjacent segments in the respective directions $d=1,\dots,d=4$. (b) Pattern-based rules $r_1^{\mathrm P}(v=2, P=P_1, u=1)$ and $r_2^{\mathrm P}(v=1, P=P_2, u=1)$, \ref{['eqn:rP']}, that can be used for the generation of images with black and white checkerboard patterns.
• Figure 3: for the generation of $3 \times 3$ images with checkerboard patterns defined by the rules from \ref{['fig:patterns2d']}. In each iteration $k \in [1,9]$, three steps take place: (i) a segment identifier $s(k)$ with the smallest entropy is drawn, (ii) a corresponding value $v_{s(k)}$ is drawn according to the pattern-based ruleset, and (iii) the newly generated identifier-value pair is added as a segment to the content instance $C(k)$. For $s(k)$ and $v(k)$, the uniformly distributed set of possible choices are listed. The choices for $s(k)$ depend on the Shannon entropy $H_k$ of each undefined segment (see appendix), as shown on the right. The choices for $v(k)$ depend on the fulfillment of the two rules $r_1^{\mathrm P}$ and $r_2^{\mathrm P}$, as listed.
• Figure 4: circuit with $\sigma = (1,\dots,N)$. (a) Circuit layout with an exemplary alphabet $A$ with $W=16$ symbols, which requires a set $\mathcal{Q}_i$ of $q=4$ qubits for each segment with $i \in [1,N]$. In each iteration $k \in [1,N]$, the operator $U_k$ prepares the qubits from $\mathcal{Q}_k$ in a superposition state $\ket{C'}_{k}$, entangled with (a subset of) the qubits from $\mathcal{Q}_1,\dots,\mathcal{Q}_{k-1}$. All qubits are measured to obtain $p(C)$ (see appendix). (b) Gate decomposition of $U_k$ into the components of $U_k^{\mathrm{coin}}$, which consists of pairs $U_k^{\mathrm{ctrl}}(C') \otimes U_k^{\mathrm{init}}(C')$. Each of these pairs represent a conditional loading of a probability distribution. The combined control operators act on the qubits from $\mathcal{Q}_k^+$, whereas the initialization operators act on the qubits from $\mathcal{Q}_k$ (see appendix).
• Figure 5: for the generation of $3 \times 3$ images with black and white checkerboard patterns in analogy to \ref{['fig:cgeneration2d']}. (a) Predefined segment order $\sigma = (1,2,3,6,5,4,7,8,9)$. (b) Circuit layout with the same symbols as in \ref{['fig:qwfccircuit']}. (c) Generated content instances $C_1$ and $C_2$ with $p(C_1)=p(C_2)=\frac{1}{2}$.