Randomized benchmarking for qudit Clifford gates

In this subsection, we describe generalization of Pauli group for qudits. First, we begin with the mathematical concept of qudits and introducing generalized Pauli operations. Then we explain generalized Pauli group. Finally, we describe this group for n qudits.

Mathematically, a qudit is a vector in d-dimensional Hilbert space ${\mathcal{H}}_{d}\cong {\mathbb{C}}^{d}$ spanned by the orthonormal computational basis $\left\{\vert s\rangle ;s\in {\mathbb{Z}}_{d}\right\}$ where ${\mathbb{Z}}_{d}{:=}\left\{0,1,\dots ,\;d-1\right\}$. Qudit unitary transformations are represented by unitary matrices {U_d } ∈ U(d), with U(d) the d-dimensional unitary Lie group. These unitary transformations include generalized Pauli operations Ξ_d , namely,

$$\begin{equation}{X}_{d}\vert s\rangle =\vert s\oplus 1\rangle ,\quad {Z}_{d}\vert s\rangle ={\omega }^{s}\vert s\rangle ,\quad \omega {:=}\mathrm{exp}\left(2\pi \text{i}/d\right)\end{equation} \tag{ 2 }$$

being defined by their actions on computational basis states with ⊕ denoting addition modulo d [1, 2]. Both X_d and Z_d have order d, obeying ${X}^{d}={Z}^{d}={\mathbb{1}}_{d}$. The usual Pauli operators arise for d = 2.

The generalized Pauli operators X_d and Z_d satisfy the commutation relation

$$\begin{equation}{Z}_{d}{X}_{d}=\omega {X}_{d}{Z}_{d}.\end{equation} \tag{ 3 }$$

These operators generate the generalized Pauli group

$$\begin{equation}{\mathcal{P}}_{d}{:=}\langle \tilde {\omega }{\mathbb{1}}_{d},{X}_{d},{Z}_{d}\rangle ,\end{equation} \tag{ 4 }$$

for $\tilde {\omega }=\omega$ if d is odd and $\tilde {\omega }={\omega }^{1/2}$ if d is even. If d is odd, the order of Z_d X_d is d, whereas, if d is even, then the order of Z_d X_d is 2d, which contributes additional roots of unity. Therefore, $\tilde {\omega }$ is defined differently for d even vs odd. The difference between the generalized and usual Pauli operators is that the qudit operators for d > 2 are only unitary, and not Hermitian. Hence, any eigenvalue of a qudit Pauli operator can be estimated only via quantum phase estimation algorithm [26].

The n-qudit Hilbert space and the associated state (density operator) space are denoted ${\mathcal{H}}_{d}^{ \otimes n}$ and $\mathcal{D}\left({\mathcal{H}}_{d}^{ \otimes n}\right)$, respectively. For n qudits, the generalized Pauli group is ${\mathcal{P}}_{d}^{n}{:=}{\mathcal{P}}_{d}^{\;\otimes \;n}$. This generalized Pauli group further generalizes to ${\tilde {\mathcal{P}}}_{d}^{n}{:=}{\mathcal{P}}_{d}^{n}/\langle \tilde {\omega }{\mathbb{1}}_{d}\rangle$ without phases for $\langle \tilde {\omega }{\mathbb{1}}_{d}\rangle$, which indicates the group generated by $\tilde {\omega }{\mathbb{1}}_{d}$ [38].

In this subsection, we explain n-qudit Clifford group. First we begin the section by describing the Clifford group for qudits. Then we explain the simple case of d = 3.

As the normalizer of the generalized Pauli group, the qudit Clifford group comprises all unitary operators that map ${\mathcal{P}}_{d}^{n}$ to itself under conjugation. Hence, the n-qudit Clifford group (5) is [38]

$$\begin{align}\hfill {\mathcal{C}}_{d}^{n}{:=}& \left\{c\in U\left({d}^{n}\right);c{\tilde {\mathcal{P}}}_{d}^{n}{c}^{{\dagger}}\in {\mathcal{P}}_{d}^{n}\right\}/\left\{{\text{e}}^{\text{i}\theta }{\mathbb{1}}_{d}^{n};\;\theta \in \mathbb{R}\right\}\hfill \\ \hfill =& \,\langle {\text{CZ}}_{d},{F}_{d},{P}_{d},{Z}_{d}\rangle ,\hfill \end{align} \tag{ 5 }$$

i.e., generated by controlled-Z

$$\begin{equation}\mathrm{C}\mathrm{Z}\vert s{s}^{\prime }\rangle {:=}{\omega }^{s{s}^{\prime }}\vert s{s}^{\prime }\rangle ,\end{equation} \tag{ 6 }$$

quantum Fourier transform

$$\begin{equation}F\vert s\rangle {:=}\frac{1}{\sqrt{d}}\sum _{{s}^{\prime }\in {\mathbb{Z}}_{d}}{\omega }^{s{s}^{\prime }}\vert {s}^{\prime }\rangle ,\end{equation} \tag{ 7 }$$

phase gate

$$\begin{equation}P\vert s\rangle {:=}{\omega }^{\frac{s\left(s+{\varrho }_{d}\right)}{2}}\vert s\rangle ,\quad {\varrho }_{d}=\begin{cases}1,\quad \hfill & \text{if}\;d\;\text{is}\;\text{odd,}\hfill \\ 0,\quad \hfill & \text{otherwise,}\hfill \end{cases}\end{equation} \tag{ 8 }$$

and Pauli-Z gates [39–41]. Any n-qudit Clifford gate can be decomposed into multiplicative and tensor products of these gates. The cardinality of the single-qudit Clifford group has been explicitly calculated [42]. For $d\in \mathbb{P}$ (prime), the number of distinct Clifford gates (up to global phase) for the single-qudit case is ${d}^{3}\left({d}^{2}-1\right)$ [34].

For d = 3, ${\mathcal{C}}_{3}$ comprises 216 Clifford gates [43], which are generated by single-qutrit Fourier transform F₃ and phase gate S₃. Specifically, any single-qutrit Clifford operation can be obtained by a product of three elements in $\mathcal{L}$, $\mathcal{M}$ and $\mathcal{N}$ [43], where $\mathcal{L}$ is the subgroup of ${\mathcal{C}}_{3}$ generated by P₃ and X₃ and

$$\begin{equation}\mathcal{M}=\left\{\mathbb{1},{F}_{3}^{2}\right\},\quad \mathcal{N}=\left\{\mathbb{1},{F}_{3},{S}_{3}{F}_{3},{S}_{3}^{2}{F}_{3}\right\}\end{equation} \tag{ 9 }$$

for ${F}_{3}^{2}$ and ${S}_{3}^{2}$ are the squares of F₃ and S₃, respectively.

Similar to qubit Clifford circuits, quantum circuits with only prime-dimensional qudit Clifford gatescan be classically efficiently simulated [44]. To achieve universal quantum computing, at least one non-Clifford gate must be added to the set of Clifford gates. An example of single-qudit non-Clifford gates is the generalized T-gate [41]

$$\begin{equation}T\vert s\rangle {:=}{\omega }^{{s}^{3}/{d}^{2}}\vert s\rangle .\end{equation} \tag{ 10 }$$

For qubits, T-gates can be benchmarked by dihedral-group benchmarking [45], instead of benchmarking U2D, which is our focus. Whether dihedral benchmarking can be generalized to the qudit case is an open problem.

In this subsection, we explain U2D, which is an essential concept for a scalable and efficient URB. Then we discuss U2D for the multi-qudit Clifford group.

A U2D comprises unitary matrices ${\left\{{U}_{j}\right\}}_{j=1}^{K}$ satisfying [37]

$$\begin{equation}\frac{1}{K}\sum _{j=1}^{K}{U}_{j}^{{\dagger}}\mathcal{E}\left({U}_{j}\rho {U}_{j}^{{\dagger}}\right){U}_{j}={\int }_{U\left(d\right)}\mathrm{d}U\;{U}^{{\dagger}}\mathcal{E}\left(U\rho {U}^{{\dagger}}\right)U,\end{equation} \tag{ 11 }$$

for any quantum channel $\mathcal{E}$ [46] and any state ρ with dU denoting the unitarily invariant Haar measure [47] on the Lie group U(d). Equation (11) implies that twirling any quantum channel over U2D is equivalent to twirling over a Haar-measure unitary group. The multi-qubit Clifford group ${\mathcal{C}}_{2}^{n}$ for n qubits forms a U2D [37, 48].

Though not explicitly stated, the multi-qudit Clifford group ${\mathcal{C}}_{d}^{n}$ evidently forms a U2D based on Webb's analysis of Pauli-mixing Clifford ensembles [38]. A set $\mathcal{S}\subseteq {\mathcal{C}}_{d}^{n}$ is called a Pauli-mixing Clifford ensemble (definition 3 in [38]) if every pair of non-identity Pauli operators are related, up to a phase, by a Clifford conjugation and the number of Clifford operators for each conjugation is constant. Thus, a Clifford operator chosen uniformly at random from $\mathcal{S}$ maps every non-identity Pauli operator to every other non-identity Pauli operator with equal probability. Webb's lemma 3, together with lemma 2, says that ${\mathcal{C}}_{d}^{n}$ is both Pauli-invariant (for each $c\in \mathcal{S}$, cΞ, up to a phase, is in $\mathcal{S}$ as well for all ${\Xi}\in \mathcal{P}$) and Pauli-mixing, and lemma 1 says that, if a Pauli-invariant Clifford ensemble is Pauli-mixing, then this ensemble is a U2D. Hence, ${\mathcal{C}}_{d}^{n}$ is a U2D.

In this subsection we show how exploiting U2D property of n-qudit Clifford group result in depolarizing channel. Twirling a quantum channel with respect to a group of unitary operations is the basic approach utilized by URB. Twirling $\mathcal{E}$ with average fidelity

$$\begin{equation}{\overline{F}}_{\mathcal{E}}={\int }_{{\mathcal{H}}_{d}}\mathrm{d}\phi \langle \phi \vert \mathcal{E}\left(\vert \phi \rangle \langle \phi \vert \right)\vert \phi \rangle \end{equation} \tag{ 12 }$$

over Haar-random unitary operations yields a depolarizing channel [49, 50]

$$\begin{equation}{\int }_{U\left(d\right)}\mathrm{d}U\;{U}^{{\dagger}}\mathcal{E}\left(U\rho {U}^{{\dagger}}\right)U={\mathcal{E}}_{\text{dep}}\left(\rho \right),\end{equation} \tag{ 13 }$$

with the same average fidelity as for $\mathcal{E}$, where

$$\begin{equation}{\mathcal{E}}_{\text{dep}}\left(\rho \right){:=}p\rho +\left(1-p\right)\frac{\mathbb{1}}{{d}^{n}},\quad p=\frac{{d}^{n}{\overline{F}}_{\mathcal{E}}-1}{{d}^{n}-1}.\end{equation} \tag{ 14 }$$

Proposition 1. Twirling a channel over an n-qudit Clifford group yields a depolarizing channel.

Proof. Combining equation (13) with the fact that the n-qudit Clifford group forms a U2D yields

$$\begin{equation}\frac{1}{\vert {\mathcal{C}}_{d}^{n}\vert }\sum _{l=1}^{\vert {\mathcal{C}}_{d}^{n}\vert }{\mathcal{C}}_{l}^{-1}{\circ}\mathcal{E}{\circ}{\mathcal{C}}_{l}={\mathcal{E}}_{\text{dep}}.\end{equation} \tag{ 15 }$$

Averaging over the n-qudit Clifford group is identical to averaging over the unitary group with respect to the uniform Haar measure so equation (15) follows. □

In this subsection, first, we set some notation that will be used throughout. Then we discuss how to generate a random sequence of Clifford gates, and obtain averaged sequence fidelity.

A concatenation, or composition, of n-qudit Clifford gates in a sequence of length m is denoted ${{\unicode{9711}}}_{j=1}^{m}{\mathcal{C}}_{{i}_{j}}\mathrm{f}\mathrm{o}\mathrm{r}{\unicode{9711}}$ indicating concatenation. We use i_j as labels for the jth element in the ith sequence, where ${i}_{j}\in \left[\left\vert {\mathcal{C}}_{d}^{n}\right\vert \right]$, with [k] := {1, ..., k}, and the gate is denoted ${\mathcal{C}}_{{i}_{j}}$. All Clifford gates experience the same noise, which is represented by a noisy channel Λ following an ideal Clifford gate.

Averaging over random realizations of a sequence of Clifford gates

$$\begin{equation}{S}_{{\boldsymbol{i}}_{m}}={{\unicode{9711}}}_{j=1}^{m}{\Lambda}{\circ}{\mathcal{C}}_{{i}_{j}},\quad {\boldsymbol{i}}_{m}{:=}\left({i}_{1},{i}_{2},\dots ,\;{i}_{m}\right)\end{equation} \tag{ 16 }$$

with i _m denoting an m-tuple and ${\mathcal{C}}_{{i}_{m}}={\left({{\unicode{9711}}}_{j=1}^{m-1}{\mathcal{C}}_{{i}_{j}}\right)}^{-1}$, is equivalent to concatenating m − 1 twirled channels [22, 23]

$$\begin{equation}{{\Lambda}}_{\text{T}}{:=}\frac{1}{\vert {\mathcal{C}}_{d}^{n}\vert }\sum _{l=1}^{\vert {\mathcal{C}}_{d}^{n}\vert }{\mathcal{C}}_{l}^{-1}{\circ}{\Lambda}{\circ}{\mathcal{C}}_{l},\end{equation} \tag{ 17 }$$

followed by Λ, i.e., ${\Lambda}{\circ}{{\Lambda}}_{\text{T}}^{{\circ}m-1}$. Using proposition 1, ${{\Lambda}}_{\text{T}}^{{\circ}m}$ can be rewritten as an m-fold composition of a depolarizing channel with itself multiple times, namely,

$$\begin{equation}{{\Lambda}}_{\text{T},p}^{{\circ}m-1}\left(\rho \right)={p}^{m-1}\rho +\left(1-{p}^{m-1}\right)\mathbb{1}/{d}^{n}.\end{equation} \tag{ 18 }$$

Hence, for any input state $\vert \psi \rangle \in {\mathcal{H}}_{d}^{ \otimes n}$,

$$\begin{equation}\text{tr}\left[\vert \psi \rangle \langle \psi \vert {\Lambda}{\circ}{{\Lambda}}_{\text{T},p}^{{\circ}m-1}\left(\vert \psi \rangle \langle \psi \vert \right)\right]\end{equation} \tag{ 19 }$$

is channel fidelity averaged over random realizations of the sequence.

In this subsection, we present the fitting function for URB, by which we model the behaviour of averaged sequence fidelity. In practice, with quantum noise, equation (19) is replaced by

$$\begin{equation}{F}_{\text{seq}}\left(m\right){:=}\text{tr}\left[{E}_{\psi }{\Lambda}{\circ}{{\Lambda}}_{\text{T},p}^{{\circ}m-1}\left({\rho }_{\psi }\right)\right]\end{equation} \tag{ 20 }$$

for E_ψ and ρ_ψ the positive-operator valued measure (POVM) [26] element and quantum state including SPAM errors, respectively. Plugging equation (18) into equation (20) yields

$$\begin{equation}{F}_{\text{seq}}\left(m\right)={A}_{0}{p}^{m-1}+{B}_{0},\end{equation} \tag{ 21 }$$

which absorbs SPAM errors, for

$$\begin{equation*}{A}_{0}{:=}\text{tr}\left[{E}_{\psi }{\Lambda}\left({\rho }_{\psi }-\frac{\mathbb{1}}{{d}^{n}}\right)\right],\quad {B}_{0}{:=}\frac{\text{tr}\left[{E}_{\psi }{\Lambda}\left(\mathbb{1}\right)\right]}{{d}^{n}}\end{equation*}$$

being the coefficients.

In this section, exploiting U2D property of n-qudit Clifford group we presented twirling a channel over this group yields a depolarizing channel. Then, we obtained averaged sequence fidelity from depolarizing rate. Furthermore, we introduced fitting model for our URB. This section relies on the assumption that quantum noise is gate-independent. As gate-dependent noise decays in the same form as gate-independent noise plus a perturbation [51], our approach could be naturally extended to the case of gate-dependent noise.

In this subsection, we provide pseudocode for our multi-qudit URB procedure, which is immensely useful to ensure that the procedure flows logically and does not leave out any key steps. Our algorithm is designed to estimate average gate fidelity ${\overline{F}}_{{\Lambda}}$ over a Haar-random set of input states.

Our pseudocode uses the following data types, expressed conventionally as all capitals. UNSIGNED INTEGER refers to a positive integer in ${\mathbb{Z}}^{+}$, and REAL, COMPLEX, BINARY, DARY and INTERVAL refer to real $\mathbb{R}$, complex $\mathbb{C}$, binary {0, 1}, d-ary {0, 1, ..., d − 1} and the unit interval $\left[0,1\right]\subset \mathbb{R}$, respectively. Besides classical data types, we introduce quantum data types as well [52]. QDARY refers to a qudit of dimension d. Each of these data types can be an array with data type followed by brackets [ ]; a sequence of two brackets [ ][ ] denotes a two-dimensional array, which is readily generalized to higher-dimensional arrays by adding more brackets.

For pseudocode variables, we use camelCase, so qudit number n is denoted by numQud and of type UNSIGNED INTEGER, and Hilbert-space dimension d is HilbDim and also of type UNSIGNED INTEGER. We use numSeq to denote the maximum number of different gate sequences of a fixed length and of type UNSIGNED INTEGER. QDARY[ ] indicates a multi-qudit state and QDOP is an operation that maps a QDARY[HilbDim]-typed variable to another QDARY[HilbDim]-typed variable.

In describing our algorithm using pseudocode, we employ functions from an ideal library explained here. We use rand(maxInt) to generate a uniformly random integer in [maxInt]. prod maps two Clifford-gate indices to the index corresponding to the product of these two referenced Clifford gates. inv maps one Clifford-gate index to the index corresponding to the inverse of the Clifford gate. We use data type prep for preparing a qudit pure state according to a classical description of the state, and projMeas denotes qudit-state measurement that yields 1 if the qudit state is projected onto a certain pure state and otherwise yields 0. For statistical processes, we employ avg, which calculates the average of all entries in an array, and fit, which is a least-squares regression algorithm. Our randomized benchmarking algorithm comprises input, output and procedure, which we now describe in plain English.

4.1.1. Input

4.1.2. Output

The output of the algorithm is an estimate $\hat{r}$ of the average gate infidelity, often called average error rate,

$$\begin{equation}r{:=}1-{\overline{F}}_{{\Lambda}}\in \left[0,1\right].\end{equation} \tag{ 22 }$$

This estimate is a URB figure of merit that characterizes average performance of qudit Clifford gates.

4.1.3. Procedure

Now we explain the URB procedure for qudit Clifford gates. We initialize the n-qudit state |ψ⟩ as the pure state ${\left\vert 0\right\rangle }^{\;\otimes n}$. Then we generate k random sequences of n-qudit Clifford gates ${\mathcal{C}}_{{i}_{j}}$ each of length j, where 2 ⩽ j ⩽ m, as k samples of a random sequence. The first j − 1 gates in each sequence are uniformly randomly chosen from ${\mathcal{C}}_{d}^{n}$, and the final gate is determined by the first j − 1 gates according to

$$\begin{equation}{\mathcal{C}}_{{i}_{j}}={\left({{\unicode{9711}}}_{{j}^{\prime }=1}^{j-1}{\mathcal{C}}_{{i}_{{j}^{\prime }}}\right)}^{-1}.\end{equation} \tag{ 23 }$$

As Clifford gates form a group, this final gate is also an element of the group.

We apply each of the k sequences of qudit gates to the initial state. Then we apply measurements corresponding to the POVM $\left\{\left\vert \psi \right\rangle \langle \psi \vert ,\mathbb{1}-\left\vert \psi \right\rangle \left\langle \psi \right\vert \right\}$ on the output state. If the measurement outcome corresponds to |ψ⟩⟨ψ|, we assign a value of one, otherwise, a value of zero. By averaging over k different sequences and l copies of each sequence, we obtain an estimate $\hat{F}\left(j\right)$ of the averaged sequence fidelity

$$\begin{equation}{F}_{\text{seq}}\left(j\right)=\text{tr}\left({E}_{\psi }{\mathcal{S}}_{j}\left({\rho }_{\psi }\right)\right),\quad {\mathcal{S}}_{j}=\frac{1}{k}\sum _{{\boldsymbol{i}}_{\boldsymbol{j}-1}\in {\left[\left\vert {\mathcal{C}}_{d}^{n}\right\vert \right]}^{\;\otimes \;j-1}}{S}_{{\boldsymbol{i}}_{j}}.\end{equation} \tag{ 24 }$$

Now we repeat the above procedure for different values of j, which increases from two to the maximum length m in succession. Finally, we fit the estimates $\hat{F}\left(j\right)$ to equation (21) with p the decay parameter and $\hat{p}$ its estimate.

Per equation (14), we see that fidelity decay parameter p is related to r via

$$\begin{equation}r=\left(1-p\right)\left(1-\frac{1}{{d}^{n}}\right).\end{equation} \tag{ 25 }$$

Therefore, by estimating p from URB of Clifford gates, we obtain the output average infidelity (algorithm 1).

Algorithm 1.  Randomized benchmarking.

Input:
UNSIGNED INTEGER numQud	${\unicode{9657}}$ # qudits
UNSIGNED INTEGER HilbDim	${\unicode{9657}}$ Hilbert-space dimension
UNSIGNED INTEGER cardCliff	${\unicode{9657}}$ Cardinality of numQud-qudit Clifford group
UNSIGNED INTEGER gateIndex[cardCliff]	${\unicode{9657}}$ Array of multi-qudit gates' labels
UNSIGNED INTEGER maxLeng	${\unicode{9657}}$ Maximal sequence length
UNSIGNED INTEGER numSeq	${\unicode{9657}}$ # sequences for each length
UNSIGNED INTEGER numCop	${\unicode{9657}}$ # copies for each gate sequence
Output:
INTERVAL avInfid	${\unicode{9657}}$ Average infidelity
1: procedure randBench(numQud, HilbDim, CliffGate, maxLeng, numSeq, numCop)
2:  COMPLEX initState[HilbDim ∧ numQud]←[0,0,...,0];	${\unicode{9657}}$ The initial state is the numQud-qudit pure state
3:  REAL estRegr[3];	${\unicode{9657}}$ Base, slope and intercept for exponential fiting
4:  REAL seqFid[maxLeng];	${\unicode{9657}}$ Fidelity averaged over gate sequences
5:  UNSIGNED INTEGER currIndex;
6:  QDARY state[numQud];
7:  for k = 2 : maxLeng do
8:   BINARY outcome[numCop];	${\unicode{9657}}$ Array of measurement outcomes
9:   REAL survProb[numSeq];	${\unicode{9657}}$ Array of survival probabilities of the initial state
10:    for i = 1 : numSeq do
11:    currIndex ← 1;	${\unicode{9657}}$Overwrite currIndex with 1, the index referred to indentity channel
12:    for j = 1 : k – 1 do
13:     gateIndex[j]←rand(cardCliff);	${\unicode{9657}}$ Generate a random integer from one to cardCliff.
14:     currIndex ← prod(currIndex, gateIndex);
15:    end for
16:    gateIndex[k] ← inv(currIndex);
17:    for l = 1 : numCop do
18:     state ← prep(initState);
19:     for j = 1 : k do
20:      QDOP CliffGate[cardCliff];	${\unicode{9657}}$ Array of multi-qudit gates
21:      state ← CliffGate[gateIndex[j]] * state;	${\unicode{9657}}$ jth Clifford gate maps old state to new state
22:     end for
23:     if projMeas(state, initState) = 1 then
24:        outcome[l] ← 1;	${\unicode{9657}}$ Assign 1 if each single-qudit state is projected onto initState
25:     else
26:       outcome[l] ← 0;	${\unicode{9657}}$ Otherwise, assign 0
27:     end if
28:    end for
29:    survProb[i] ← avg(outcome);
30:   end for
31:   seqFid[k] ← avg(survProb);
32:  end for
33:  estRegr ← fit(seqFid);	${\unicode{9657}}$ Least-squares regression algorithm with exponential fitting model
	y = estRegr[2] ∗ estRegr[1] ∧ x + estRegr[3].
34:  return avInfid ← (1 − estRegr[1])* (1 − 1 / HilbDim ∧ numQud).
35: end procedure

Now we explain why previous work on multi-qubit URB does not readily yield qudit URB. One might expect that solving multi-qubit URB would yield qudit URB trivially. Such an approach would exploit Schur–Weyl duality [53]. Schur–Weyl duality, applied to the symmetry group S_n and the unitary group U(d), which have commuting actions on the n-fold tensor product of d-dimensional Hilbert spaces, ${\mathcal{H}}_{d}^{ \otimes n}$, states that, under the joint action of S_n and U(d), the tensor product space decomposes into a direct sum of tensor products of irreducible modules. The question is whether we can use that to construct Clifford operators for qudits. We show that this enticing notion is fallacious by falsifying the following proposal.

Proposal 1. A tensor product of any n single-qubit Clifford operators is a direct sum of a Clifford operator for the (n + 1)-dimensional symmetric space with any operators for the remaining partially and antisymmetric spaces (FALSE).

This proposal is enticing because we could simply use existing multi-qubit benchmarking work [22] instead of producing a new result.

Mathematically, this proposal can be expressed as follows. Let ${\left\{{C}_{{\imath}}^{\left(2\right)}\right\}}_{{\imath}=1}^{n}$ be a sequence of n single-qubit operators, and let C⁽ⁿ⁺¹⁾ be any Clifford operator on an (n + 1)-dimensional Hilbert space ${\mathcal{H}}_{d}$ for d = n + 1. The conjecture is then that, for all ${\left\{{C}_{{\imath}}^{\left(2\right)}\right\}}_{{\imath}=1}^{n}$, a Clifford operator C⁽ⁿ⁺¹⁾ exists such that

$$\begin{equation}\underset{{\imath}}{\bigotimes}{C}_{{\imath}}^{\left(2\right)}={C}^{\left(n+1\right)}\oplus {{\Theta}}^{\left({2}^{n}-n-1\right)},\end{equation} \tag{ 26 }$$

for any (2ⁿ − n − 1)-dimensional unitary operator ${{\Theta}}^{\left({2}^{n}-n-1\right)}$. We now demonstrate that this proposal is false by giving a counterexample.

Counterexample. This proposal is falsified with a counterexample, specifically for ${C}_{{\imath}}^{\left(2\right)}=H$ for ı ∈ {1, 2}. The two-qubit Hadamard gate is

$$\begin{equation}H\;\otimes \;H=R\oplus \left(-1\right),\quad R{:=}\frac{1}{2}\left(\begin{pmatrix}\hfill 1\hfill & \hfill \sqrt{2}\hfill & \hfill 1\hfill \\ \hfill \sqrt{2}\hfill & \hfill 0\hfill & \hfill -\sqrt{2}\hfill \\ \hfill 1\hfill & \hfill -\sqrt{2}\hfill & \hfill 1\hfill \end{pmatrix}\right),\end{equation} \tag{ 27 }$$

which is block-diagonal on both the symmetric and anti-symmetric subspaces. However, the block part R on the three-dimensional symmetric subspace is not a qutrit Clifford gate, as $R{X}_{3}R\notin {\mathcal{P}}_{3}$. □

Therefore, the proposal is falsified: a tensor product of qubit Clifford gates cannot in general be written as a direct sum involving Clifford gates, and qudit Clifford gates are not directly obtained from multi-qubit Clifford gates over the symmetric subspace. This falsification implies a significant difference between quantum computing on qudits vs on multiple qubits, even for the same total dimension.