Strongly Entangling Layer (SEL)

• SEL is a quantum circuit subroutine that maximizes entanglement by using specialized two-qudit gates to achieve the largest entropy increase.
• Its design leverages rigorous criteria like Cohen’s theorem and group expansion to ensure a maximally entangling capacity across bipartitions.
• Practical SEL implementation balances optimal entanglement generation with hardware constraints, including connectivity, ancilla allocation, and noise tradeoffs.

A Strongly Entangling Layer (SEL) is a quantum circuit subroutine whose purpose is to maximize the entangling capacity across designated bipartitions of a multi-qudit register. SELs are constructed from two-qudit gates that are themselves maximally entangling: they achieve the largest possible increase in entropy of entanglement when applied to appropriately chosen product–ancilla inputs. The structure of SELs, their mathematical criteria, and their implementation tradeoffs form a foundation for maximizing entanglement generation in quantum computation and quantum information processing (Cohen, 2011).

1. Maximally Entangling Unitaries and Entangling Capacity

Let $$U:\mathcal{H}_A\otimes\mathcal{H}_B \to \mathcal{H}_A\otimes\mathcal{H}_B$$ be a bipartite unitary operator, where Alice and Bob each control systems $A$ (dimension $d_A$) and $B$ (dimension $d_B\ge d_A$) respectively. Ancillary systems $a$ (Alice) and $b$ (Bob) of dimensions $d_a$ and $d_b$ may be appended, with $d_a=d_A$, $A$0 providing optimality.

The entangling capacity of $A$1 is defined as

$A$2

where $A$3, and $A$4 is the entropy of entanglement across the $A$5 partition.

The maximal entangling capacity is known to satisfy $A$6, achievable if and only if $A$7 is maximally entangling. For maximally entangling $A$8, a canonical input achieving $A$9 is $d_A$0, where $d_A$1.

2. Cohen's Theorem: Characterization by Group Expansion

Cohen (2011) provides a necessary and sufficient condition for a bipartite unitary $d_A$2 to be maximally entangling:

Let $d_A$3 be any finite group of order $d_A$4 with an irreducible (possibly projective) representation $d_A$5 on $d_A$6. Any unitary $d_A$7 can be expanded as

$d_A$8

where $d_A$9 may be chosen as generalized Pauli operators $B$0 with $B$1.

Theorem 1: $B$2 is maximally entangling if and only if there exists a positive semi-definite operator $B$3 on $B$4 such that, for all $B$5,

$B$6

The optimal ancilla input state on $B$7 is the purification of $B$8.

Equivalently, in the operator Schmidt decomposition, $B$9 must have Schmidt rank $d_B\ge d_A$0, and the Schmidt coefficients $d_B\ge d_A$1 must all be $d_B\ge d_A$2: $d_B\ge d_A$3

3. Practical Verification of Maximally Entangling Gates

The process to verify if a given two-qudit gate $d_B\ge d_A$4 is maximally entangling is as follows:

1. Expand $d_B\ge d_A$5 in the generalized Pauli basis on $d_B\ge d_A$6:

$d_B\ge d_A$7

for each $d_B\ge d_A$8.

1. Compute the Gram matrix $d_B\ge d_A$9 for some positive $a$0. Search for $a$1 such that $a$2.
2. Concretely, stack the matrix elements of $a$3 and determine the null space for $a$4; any positive $a$5 normalizing the diagonal to $a$6 attests to maximal entangling power.

If no such $a$7 can be found, $a$8 is not maximally entangling.

4. Construction and Graph-Theoretic Properties of SELs

A Strongly Entangling Layer on $a$9 qudits is defined as a parallel layer of two-qudit maximally entangling gates $b$0 such that for every target bipartition $b$1 of the register, the composite operator

$b$2

is itself maximally entangling across the $b$3 cut. Here, $b$4 is the interaction graph imposed by hardware connectivity.

Guidelines for constructing SELs:

• The graph $b$5 must connect every bipartition; that is, every cut $b$6 must be crossed by at least one edge $b$7, requiring $b$8 to be an expander.
• On each edge, choose $b$9 from the class of maximally entangling gates for $d_a$0.
• The global entanglement achievable across any bipartition is up to $d_a$1 ebits, and this is saturated by ensuring every qudit in the smaller partition connects via a maximally entangling gate to the complement.

A SEL thus consists of a perfect matching or generalized expander subgraph of maximally entangling gates that intersects every bipartition.

5. Explicit Families in Low Dimensions

For $d_a$2 (“qubit-qubit”), any two-qubit unitary can, up to local unitaries, be presented in Cartan form: $d_a$3 $d_a$4 is maximally entangling if and only if two of the three angles satisfy $d_a$5 for $d_a$6.

Canonical maximally entangling gates include:

Gate	Cartan Parameters	Notes
Double-CNOT	$d_a$7, $d_a$8
SWAP	$d_a$9

For $d_b$0, $d_b$1 (“qubit–qutrit”), take $d_b$2 and $d_b$3 as Pauli matrices. A maximally entangling example is the controlled-shift: $d_b$4 where $d_b$5. The Gram-matrix test confirms $d_b$6 ebits.

6. Practical Constraints and Optimization in Hardware

Considerations intrinsic to physical realization:

• Connectivity graph: On 1D chains, covering all bipartitions with maximally entangling edges necessitates circuit depth $d_b$7. For 2D lattices, SELs can be implemented in constant depth using two-round matchings.
• Ancilla allocation: Ancillas of sizes $d_b$8 and $d_b$9 are necessary in principle, though for $d_a=d_A$0 the Bob’s ancilla may be embedded in $d_a=d_A$1 itself, eliminating the need for an explicit ancilla $d_a=d_A$2.
• Parameter tradeoffs: The continuous parameter freedom in maximally entangling gate families (e.g., the third Cartan angle in $d_a=d_A$3) allows for selection based on gate duration and error rates in a given platform.
• Entanglement vs. noise: Maximally entangling gates have increased duration and are more susceptible to noise. Submaximal gates (e.g., a single CNOT, achieving 1 ebit out of a possible 2) may be preferable under certain error models, possibly requiring increased circuit depth.
• Gate compilation: Local single-qudit unitaries are used to convert a target two-qudit gate into canonical maximally entangling form, focusing the physical implementation on the nonlocal parameters.

7. Summary and Significance

Cohen's group-expansion theorem delivers a stringent, testable criterion (Equation 2.1) for a gate to be maximally entangling (Cohen, 2011). The linear algebraic structure streamlines verification and explicit construction in physical models. SELs, as layers of such gates crossing all bipartitions of interest, underlie circuit layouts seeking to saturate the entangling potential of quantum hardware for benchmarking, quantum simulation, and information scrambling tasks. Practical deployment requires balancing theoretical entangling capacity against hardware-imposed tradeoffs such as connectivity, depth, noise, and ancilla overhead.