Approximate stabilizer rank and improved weak simulation of Clifford-dominated circuits for qudits
Bravyi and Gosset recently gave classical simulation algorithms for quantum circuits dominated by Clifford operations. These algorithms scale exponentially with the number of -gates in the circuit, but polynomially in the number of qubits and Clifford operations. Here we extend their algorithm to qudits of odd prime dimension. We generalize their approximate stabilizer rank method for weak simulation to qudits and obtain the scaling of the approximate stabilizer rank with the number of single-qudit magic states. We also relate the canonical form of qudit stabilizer states to Gauss sum evaluations and give an algorithm for calculating the inner product of two -qudit stabilizer states.
With the prospect of noisy intermediate scale quantum (NISQ) computers with qubits appearing in the next decade [4, 30], determining the minimal classical cost of simulation of quantum computers has received much recent attention [8, 35, 18, 5, 29].
The Gottesman-Knill theorem shows that Clifford circuits are efficiently classically simulatable . Adding any non-Clifford gate creates a universal gate set 111An elegant recent presentation of this result in group-theoretic terms is given in [nebe2001invariants] and is briefly summarized in . One such choice for a non-Clifford gate is the gate: . Bravyi and Gosset gave a classical algorithm for simulation of quantum circuits that scales exponentially with the number of -gates in the circuit but polynomially with the number of qubits and Clifford gates . This algorithm was further developed in .
What is supplied by the addition of -gates to a Clifford circuit? The fault tolerant implementation of Clifford+ circuits substitutes magic states for each gate [9, 40]. Colloquially, gates add “magic” to a Clifford circuit. Magic is supplied by contextuality, a longstanding source of puzzles and paradoxes in the foundations of quantum mechanics .
The relationship of magic to contextuality also provides a connection to quasiprobability representations of quantum mechanics [36, 13, 13]. Specifically, positivity of a quasiprobability representation is equivalent to the absence of contextuality, and such positive states, operations and measurements admit efficient classical simulation in some cases [38, 28]. Classical statistical theories with an imposed uncertainty principle can reproduce these positive quasiprobabilistic theories for Gaussian states and qudits with [37, 3].
Pashayan et al. gave an algorithm allowing a positive quasiprobability description to include some negativity . Comparing the algorithms of Bravyi and Gosset and Pashayan should shed more light on the relationship between magic, contextuality and negativity [8, 34]. However quasiprobability representations for qubits are distinct from their -dimensional cousins [24, 25, 26]. The desire to understand the relationship between magic, contextuality and negativity therefore motivates extension of the algorithm of Bravyi and Gosset to qudits with dimension greater than two. In the present paper we extend the algorithm of Bravyi and Gosset to qudits of odd prime dimension.
The structure of the paper is as follows. In Sections II and III, we briefly introduce the necessary background. In Section IV we give the nonorthogonal decomposition of the magic state, and in Section V we give results on approximate stabilizer rank and weak simulation algorithm for qudits. We close the paper by briefly comparing our algorithm to that of .
Ii Qudit Pauli group and Clifford gates
The Pauli and Clifford groups were first generalized beyond qubits by Gottesman . Assuming henceforth that is an odd prime, we define the Heisenberg-Weyl operators:
where , where denotes addition modulo , , , where and are integers modulo , and . The Heisenberg-Weyl operators form a group whose product rule follows from the Heisenberg-Weyl commutation relation :
where is the symplectic inner product: .
The generators of the Clifford group on qudits are , and , where , and . We can also write any single qudit Clifford unitary as , where and is a matrix with entries modulo . We will make particular use of matrices for . The order of is . The Clifford group is reviewed in more detail in Appendix A.
Qudit stabilizer states can be prepared from a logical basis state by a qudit Clifford circuit. The Gottesmann-Knill theorem generalizes to qudits and qudit stabilizer computations allow efficient classical simulation . Qudit stabilizer states possess canonical forms in the logical basis just as in the qubit case [31, 12, 19].
The remaining generalization we require is an efficient classical algorithm for obtaining the inner product of two stabilizer states. This is required by the algorithm of Bravyi and Gosset and the qubit case was given in . We give an algorithm for the inner product of two -qudit stabilizer states based on Gauss sums in Appendix F.
The qudit -gate was defined in [22, 11] as a diagonal gate that maps Pauli operators to Clifford operators. Its action is specified by the image of under . Magic states are then eigenvectors of this image. Let the eigenstate of with eigenvalue be , then the magic states are . This approach is that taken by Howard in .
The image of under can be written (up to a phase) as for , integers modulo . The effect of nonzero is simply to reorder the eigenvectors and hence we can choose . Similarly, the eigenvectors for and are related by application of , a Clifford operator. We can therefore specialize to the case and , and the gate with action:
The definition of magic states allows one to replace a Clifford+ circuit with a Clifford circuit with injected magic states [40, 9]. This construction was extended to qudits in  and we review it in Appendix D. In Section III we will review the Bravyi-Gosset algorithm for qubits which we will generalize to qudits.
Iii The Bravyi-Gosset Algorithm
Bravyi and Gosset gave algorithms for both weak and strong simulation in [8, 7]. A strong simulation outputs the probability of measuring output from a given Clifford+ circuit. A weak simulation algorithm generates samples from the probability distribution over outputs of a given Clifford+ circuit. Here we review the weak simulation algorithm. A brief summary of relevant features of the strong simulation algorithm is given in Appendix C.
The key advantage of weak simulation is that one can sample from a that is close enough to the actual . Bravyi and Gosset devised a method to approximate the -qubit magic state , where , with a superposition of stabilizer states.
The approximate stabilizer rank is defined as the minimal stabilizer rank (defined in  and reviewed in Appendix C) of a state that satisfies . A close approximation to the tensor product of magic states means a close approximation to the action of a Clifford+ circuit realized by magic state injection . Therefore, will be close enough to if is small enough.
The sampling procedure given by Bravyi and Gosset relies on standard computations of stabilizers. The extension of such computations to have long been well understood . We will therefore refer the reader to  for details of these procedures which, mutatis mutandis, can be applied in the qudit case, and focus on the approximate stabilizer rank.
We begin by reviewing the approximate stabilizer rank construction from . From the magic state defined above one can construct the equivalent magic state:
The state can be decomposed into a sum of non-orthogonal stabilizer states as follows:
where and . Then can be rewritten as
The weak simulation algorithm reduces the number of stabilizer states required by approximating . This approximation is constructed by taking a subspace of :
The stabilizer rank of this approximation state is the number of elements in , which is . The random subspace is chosen so that satisfies:
It is useful to discuss the subspaces of in the language of -ary linear codes. is a -dimensional binary linear code which can be specified by generators of length . These generators can be written in a standard form as a matrix where is the identity matrix and is a matrix. Sampling random subspaces of is therefore equivalent to sampling matrices .
Iv Nonorthogonal decompositions of qudit Magic states
The qudit magic state we want to decompose is an eigenvalue one eigenstate of the Clifford operator as defined by eq. (3). We choose a stabilizer state with non-zero inner product with the magic state and act on it with powers of to obtain stabilizer states . We know these stabilizer states are distinct because if any pair were equal then the original state would be an eigenstate of the Clifford operator and hence a magic state. The sum of these states form a decomposition of the magic state (up to a possible global phase). Because has order this state is by construction an eigenvalue one eigenstate of .
The stabilizer states in the decomposition form an orbit around the magic state. This construction was discussed previously in . There are single-qudit stabilizer states , partitioned into orbits, each orbit giving a decomposition of the magic state. Every state in each orbit has the same overlap with the magic state:
where the qudit magic state is . This property is a generalization of for the qubit case. The overlaps of the elements of the nonorthogonal basis are given by: for all s, i.e.:
This expression is that for states in a SIC-POVM, and the construction here is similar to the generation of such states from a fiducial state [14, 41]. Here we only obtain states, however. See Appendix G for the evaluation of the phase of .
The states are representatives of the orbits, each of which generated by . This is because for any , which follows simply from the action of in the logical basis. applies phases quadratic in to followed by a shift. This cannot be equal to a state generated from by any power of , which can only apply phases linear in to .
From the orbit representatives we can determine the inner product of the states in the orbit with the magic state. This is given by:
This is a cubic gauss sum which can be written:
For the case, the magnitude and phase of this cubic Gauss sum, and , are computed in Appendix E. The sum is real, although not necessarily positive. Although we do not obtain a closed form for this sum, we can compute the integer value of which maximizes its absolute value for a given . These values are tabulated for small in Table 1.
The complete form of the nonorthogonal decomposition is:
which is the generalization of eq. (5) to arbitrary .
V Weak Simulation and Approximate Stabilizer Rank
In order to get an approximation for , we can follow the method of Bravyi and Gosset for the qubit case, taking a -dimensional subspace of :
Here we label the state by , a dimensional code subspace of and is a normalization factor. Comparison with eq. (13) shows that . We require:
for a given , where the first equality follows from eq. (9) and where:
Selection of the subspace depends on two factors. First, we choose the dimension of by setting :
Note that the maximum precision that can be required from the method for given is obtained by setting , so that .
Next we find an for which is not too large. The probability of obtaining a small enough can be analyzed as in  by evaluating the expectation value of over all possible :
Here is a indicator function, i.e., it is equal to 1 when and 0 otherwise. The second equal sign stands because the expectation value of for a fixed is and
From eq. (17) we have so . Therefore from Markov’s inequality we obtain
Randomly choosing subspaces gives an such that:
and hence satisfying eq.(15), with high probability.
The upper bound for the approximate stabilizer rank of a -qudit magic state given by the above method is:
In the qubit case an explicit sum formula was given for with terms, and hence the cost of evaluating is . What is the cost of evaluating for arbitrary ? In Appendix G we give an explicit formula for as a sum of products, and hence the cost of evaluating for arbitrary is .
The motivation to study the qudit generalizations of stabilizer rank algorithms such as those in [8, 7] is to enable comparison with other simulation algorithms. In , the authors apply Monte Carlo sampling on trajectories of the quasiprobability representation to estimate the probability of a measurement outcome. They find the hardness of this strong simulation depends on the total negativity (Negativity of the inputs, gates and measurements) of the circuit. Specifically the cost of the algorithm scales with the square of the total negativity.
For Clifford+ circuits that are gadgetized so that the circuit is realized by Clifford gates with magic state injection, the negativity of the circuit only comes from the ancilla inputs of magic states. If we apply the method of  to the gadgetized circuit with an input of -qutrit magic states, the cost scales as . This result is obtained by calculating the negativity of a single-qutrit magic state.
In the present paper, we obtain a scaling of for weak simulation of qutrit Clifford+ circuits. This shows that weak simulation using the approximate rank method has superior scaling to strong simulation using the method of . A stabilizer rank based strong simulation algorithm for qudits would require new results on exact stabilizer rank of qudit magic states, a topic for future work. Recent progress in extending the qubit case has been reported in , and improvements to Pashayan’s algorithm using a discrete systems generalization of the stationary phase approximation were given in .
It should be noted that one should not think of weak simulation as easy and strong simulation as hard. The difficulty of weak and strong simulation is a property of the distribution being sampled or computed. In some cases, such as quantum supremacy, we expect the difficulty of weak and strong simulation to coincide .
If we consider negativity and stabilizer rank as two measures of quantumness, we can see that they differ. Bravyi et al.  conjectured that the magic state has the smallest stabilizer rank out of the non-stabilizer states. However, the quasi-probability of the magic state has the largest negativity. In fact, Howard and Campbell also noticed this disagreement between stabilizer rank and robustness of magic . It is worth noting the differences between stabilizer rank and approximate stabilizer rank. Namely, the approximate stabilizer rank seems to agree with other measures of quantumness such as negativity or robustness of magic in that it reaches a maxima at the magic state and a minima on stabilizer states. The exact stabilizer rank does not share these properties. This makes the investigation of the difference between exact and approximate stabilizer rank interesting.
The authors thank Robert Lemke-Oliver, Dmitris Koukoulopoulos, Juspreet Sandhu, Elizabeth Crosson, Stephen Jordan and David Gosset for helpful discussions. This work was supported by NSF award number PHY 1720395 and from Google Inc.
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Appendix A The Qudit Clifford Group
We recall that is an odd prime. In a dimensional system the Pauli operators and are defined as:
where . These operators obey the Heisenberg-Weyl commutation relation:
In dimensions the Weyl-Heisenberg displacement operators are defined by:
where ,. The qubit Pauli operators are recovered from this expression for , with , and . The Heisenberg-Weyl operators form a group with multiplication rule:
where is the symplectic inner product:
For the Weyl-Heisenberg operators are unitary but not generally Hermitian.
In the qubit case, the Clifford gates map Pauli operators to Pauli operators. In the qudit case Clifford gates map Weyl-Heisenberg operators to one another. The generators of the Clifford group are defined so that the Hadamard gate maps and the phase gate maps . The generators of the single-qubit Clifford group are:
The -dimensional Clifford operators are generated by:
We can represent the Clifford group using a matrix and a vector , both with entries in :
Specifically, a Clifford unitary is given as follows:
The multiplication rule is:
The action of the Clifford operators on the Heisenberg-Weyl operators in this representation can be given as follows:
In particular we are interested in Clifford operations defined by matrices of the form:
and we introduce the notation:
for . From Table I in Zhu  the order of any element is . Clearly , and are order . For is order and for is order .
The generators and are given by:
which follows from and and:
These expressions for and allow us to construct the and for any single qudit Clifford operation expressed as a word on the generators and .
Appendix B Qudit Magic states and gates
To go beyond Clifford group computation it is useful to introduce the Clifford hierarchy, which classifies unitary operators by their action on the Pauli group. The Clifford hierarchy was defined by Gottesman and Chuang in :
The first level of the Clifford hierarchy is the Pauli group . The Clifford group is the second level of the hierarchy, unitary operators that map the Pauli group to itself. Note that elements of the Pauli group are themselves elements of the first level of the Clifford hierarchy. The third level of the Clifford hierarchy are operators that map Pauli operators to Clifford operators. The qubit gate is such an operator because , a non-Pauli element of the second level of the Clifford hierarchy.
for . is the eigenstate of the Hadamard gate and is the eigenstate of the product of Hadamard and Phase gate .
Any magic state is equivalent as a resource to any other state obtainable from it by a Clifford operation. We can define magic states more generally as the eigenstates of Clifford operations and obtain them as follows. Taking any -type magic state , we have
where is the eigenvalue of and is a Clifford gate. This means that is the eigenstate of a new Clifford operator . The same is true for -type magic states.
Campbell et al.  used this relationship between magic states and eigenvectors of Clifford operators to extend the definition of magic states to qudits . Concurrently, equivalent extensions were obtained by Howard and Vala .
b.1 Qudit gates
Campbell et al.  define sets of gates containing all gates with the following properties:
is in the third but not the second level of the Clifford hierarchy.
Amongst this set of gates is the canonical gate
Which is defined so that it maps the operator to a Clifford operator proportional to :
Here is the multiplicative inverse of modulo . This Clifford operator has order .
This condition, and the condition , gives the following form for the (See Appendix A of ):
The parameter determines the order of the operator . For the form above is valid when . For it is valid when .
By definition maps , a generalized Pauli operator, to a non-Pauli Clifford operator and so is in the third, but not the second, level of the Clifford hierarchy. We can therefore think of as a generalized gate.
From the definition of the matrix in (45), we have for and :
for and where . The qudit version of the gate , is further generalized in , which we will discuss below.
The gate is also sometimes called the gate because
Vala and Howard developed the qudit versions of this gate concurrently with Campbell et al’s development of qudit magic states [11, 22]. The results are equivalent and we give the details of the relationship between them here.
Vala and Howard parameterize the set of diagonal gates on a single qudit as follows:
All diagonal gates fix and so their action is completely determined by . This parallels the development of Campbell et al. who considered the action of their canonical gate on the operator and insisted that the result of that action was .
Vala and Howard proceed more generally, computing the action of these diagonal matrices:
Given is diagonal, only is nontrivial.
Vala and Howard then consider the case that is in the third level of the Clifford hierarchy so that the image of can be written (c.f. eq (18) in ):
where . The right hand side here is the most general form allowed because eqn. (52) implies that the image of must be times a diagonal Clifford operator, and the most general form of a diagonal Clifford operator has and , . Combining equation (52) and (53), one obtains (c.f. eq. (19) in ):
Vala and Howard then solve for with these parameters.
This analysis is equivalent to that performed in Campbell et al. , Appendix A.
The case as usual presents some special difficulties. In the Campbell analysis one must choose for as there are no Clifford operators with , .
The set of operators for is given by:
where . The are given by:
where all operations can be taken modulo . The determinant of for can be computed from this definition:
showing that is not in for .
We can relate the diagonal operators defined by Vala and Howard and the operators defined by Campbell et. al as follows. Writing:
we wish to compare:
These are both cubic in so we can find the particular that corresponds to by equating the coefficients. We begin by setting to find the constant term. We immediately obtain:
We conclude that and will only be equivalent up to a global phase determined by this convention.
Equating the cubic terms yields . Equating the quadratic terms gives
so that . Finally, equating the linear terms gives:
We may therefore relate and for arbitrary as follows:
The first two cases of this equivalence are for amd and, up to a global phase, are as given in equations (70) and (71) of .
b.2 Qudit Magic states
The gates also allow us to find eigenstates of as follows. Define the state , where is the eigenstate of with eigenvalue . We can calculate:
Given eq.(53), Vala and Howard recovered the definition of the magic states of Campbell and showed that these magic states are eigenstates of with eigenvalue :
Appendix C Strong Simulation for qubits.
We review here the strong simulation algorithm given by Bravyi and Gosset in .
Let be the number of gates in the -qubit quantum circuit we wish to classically simulate. The first step is to replace every gate in the circuit by Clifford gates and an ancilla input of a magic state , defined in  as:
At the end of the computation we will measure of the qubits in the logical basis. This measurement with outcome (where is a bitstring of length ), postselected to the case where all ancilla measurements have result , is represented by a projector . The strong simulation algorithm classically computes the probability of this measurement outcome after acting with a Clifford circuit , which is our original (non-Clifford) circuit with all -gates replaced by the gadget of Figure 1. Therefore we can express the probability of obtaining output as:
The factor of here compensates for the fact that we postselected on the measurement outcomes of the ancilla qubits.
We define a -qubit projection operator . This projector maps states onto a stabilizer subspace. Then eq.(70) becomes
where is an integer that depends on the number of qubits we are measuring out of and the dimension of the stabilizer subspace is mapping onto.
If we can expand into a sum of stabilizer states, then we can express as a sum of inner products of -qubit stabilizer states, which can be computed in time ([1, 15, 10, 8]). The fewer stabilizer states in the expansion of , the more efficient the algorithm is.
Stabilizer rank is defined as the minimal number of stabilizer states needed to write a pure state as a linear combination of stabilizer states. The value of is trivially upper bounded by because logical basis states are stabilizer states, and is also believed to be lower bounded by an exponential in . For practical purposes we can achieve progress through a series of constructive upper bounds.
In , Bravyi et al. found a stabilizer rank upper bound by obtaining for and dividing the -qubit state into a product of -qubit states. Therefore, has a upper bound .
If we denote the stabilizer rank for the tensor product of single-qubit magic states as , the cost of classically computing by taking inner products as described above is .
The quadratic dependence on stabilizer rank can be improved by a Monte Carlo method, developed by Bravyi and Gosset, to approximate the norm of a tensor product of magic states projected on a stabilizer subspace: