Maximizing the Hilbert Space for a Finite Number of Distinguishable Quantum States

Given a particular quantum computing architecture, how might one optimize its resources to maximize its computing power? We consider quantum computers with a number of distinguishable quantum states, and entangled particles shared between those states. Hilbert-space dimensionality is linked to nonclassicality and, hence, quantum computing power. We find that qutrit-based quantum computers optimize the Hilbert-space dimensionality and so are expected to be more powerful than other qudit implementations. In going beyond qudits, we identify structures with much higher Hilbert-space dimensionalities.

Figure 1

Possible groupings of a linear array of 12 distinguishable quantum sites into various quantum qudit partitions. Qudit groupings are shown schematically as dashed ovals, and open circles are quantum sites. (a) Partitioning into six qubits, (b) partitioning into four qutrits, (c) partitioning into three 4-qudits, (d) partitioning into two 6-qudits, (e) partitioning into one 12-qudit.

Figure 2

Hilbert space dimensionality for various geometries as a function of qudit size for $N$ distinguishable quantum states. + corresponds to equipartitioning into qudits, ○ to a $Q_x(x/2,1)$ partition, × to a $Q_x(x,2)$ partition, and * to spin referenced systems. Note that the dimensionalities are defined only when $N/x$ is a positive integer, i.e., $x=2,3,4,6,12$. The lines are merely guides for the eye.

Figure 3

Possible groupings of a linear array of 12 distinguishable quantum sites into various $Q_x(k,1)$ partitions. The $x$-qudit groupings are shown schematically as dashed ovals, open circles are quantum sites, and filled circles correspond to (randomly chosen) filled sites. (a) Six $Q_2(1,1)$ partitions, (b) four $Q_3(1,1)$ partitions, (c) three $Q_4(2,1)$ partitions, (d) two $Q_6(3,1)$ partitions, and (e) one $Q_{12}(6,1)$ partition.