Static Analysis of Quantum Programs via Gottesman Types

The Heisenberg representation of quantum operators provides a powerful technique for reasoning about quantum circuits, albeit those restricted to the common (non-universal) Clifford set

H

S

and

CNOT

. The Gottesman-Knill theorem showed that we can use this representation to efficiently simulate Clifford circuits. We show that Gottesman's semantics for quantum programs can be treated as a type system, allowing us to efficiently characterize a common subset of quantum programs. We apply this primarily towards tracking entanglement in programs, showing how superdense coding and GHZ circuits entangle and disentangle qubits and how to safely dispose of ancillae. We demonstrate the efficiency of our typechecking algorithm both for simple deductions and those involving entanglement and measurement.

Related papers

Gottesman Types for Quantum Programs

Robert W. Rand, Aarthi Sundaram, Kartik Singhal +1 more

The Heisenberg representation of quantum operators provides a powerful technique for reasoning about quantum circuits, albeit those restricted to the common (non-universal) Clifford set H, S and CNOT. The Gottesman-Knill theorem showed that we can use this representation to efficiently simulate Clifford circuits. We show that Gottesman's semantics for quantum programs can be treated as a type system, allowing us to efficiently characterize a common subset of quantum programs. We also show that it can be extended beyond the Clifford set to partially characterize a broad range of programs. We apply these types to reason about separable states and the superdense coding algorithm.The Heisenberg representation of quantum operators provides a powerful technique for reasoning about quantum circuits, albeit those restricted to the common (non-universal) Clifford set H, S and CNOT. The Gottesman-Knill theorem showed that we can use this representation to efficiently simulate Clifford circuits. We show that Gottesman's semantics for quantum programs can be treated as a type system, allowing us to efficiently characterize a common subset of quantum programs. We also show that it can be extended beyond the Clifford set to partially characterize a broad range of programs. We apply these types to reason about separable states and the superdense coding algorithm.

arXiv:2109.02197Electronic Proceedings in Theoretical Computer ScienceSep 6, 202110 citations

Hoare meets Heisenberg: A Lightweight Logic for Quantum Programs

Aarthi Sundaram, Robert Rand, Kartik Singhal +2 more

We show that Gottesman's (1998) semantics for Clifford circuits based on the Heisenberg representation gives rise to a lightweight Hoare-like logic for efficiently characterizing a common subset of quantum programs. Our applications include (i) certifying whether auxiliary qubits can be safely disposed of, (ii) determining if a system is separable across a given bipartition, (iii) checking the transversality of a gate with respect to a given stabilizer code, and (iv) computing post-measurement states for computational basis measurements. Further, this logic is extended to accommodate universal quantum computing by deriving Hoare triples for the T -gate, multiply-controlled unitaries such as the Toffoli gate, and some gate injection circuits that use associated magic states. A number of interesting results emerge from this logic, including a lower bound on the number of T gates necessary to perform a multiply-controlled Z gate.We show that Gottesman's (1998) semantics for Clifford circuits based on the Heisenberg representation gives rise to a lightweight Hoare-like logic for efficiently characterizing a common subset of quantum programs. Our applications include (i) certifying whether auxiliary qubits can be safely disposed of, (ii) determining if a system is separable across a given bipartition, (iii) checking the transversality of a gate with respect to a given stabilizer code, and (iv) computing post-measurement states for computational basis measurements. Further, this logic is extended to accommodate universal quantum computing by deriving Hoare triples for the T -gate, multiply-controlled unitaries such as the Toffoli gate, and some gate injection circuits that use associated magic states. A number of interesting results emerge from this logic, including a lower bound on the number of T gates necessary to perform a multiply-controlled Z gate.

arXiv:2101.08939QuantumJul 23, 20261 citation

Classical simulation of quantum computation, the Gottesman-Knill theorem, and slightly beyond

M. Van den Nest

We study classical simulation of quantum computation, taking the Gottesman-Knill theorem as a starting point. We show how each Clifford circuit can be reduced to an equivalent, manifestly simulatable circuit (normal form). This provides a simple proof of the Gottesman-Knill theorem without resorting to stabilizer techniques. The normal form highlights why Clifford circuits have such limited computational power in spite of their high entangling power. At the same time, the normal form shows how the classical simulation of Clifford circuits fits into the standard way of embedding classical computation into the quantum circuit model. This leads to simple extensions of Clifford circuits which are classically simulatable. These circuits can be efficiently simulated by classical sampling ('weak simulation') even though the problem of exactly computing the outcomes of measurements for these circuits ('strong simulation') is proved to be #P-complete--thus showing that there is a separation between weak and strong classical simulation of quantum computation.We study classical simulation of quantum computation, taking the Gottesman-Knill theorem as a starting point. We show how each Clifford circuit can be reduced to an equivalent, manifestly simulatable circuit (normal form). This provides a simple proof of the Gottesman-Knill theorem without resorting to stabilizer techniques. The normal form highlights why Clifford circuits have such limited computational power in spite of their high entangling power. At the same time, the normal form shows how the classical simulation of Clifford circuits fits into the standard way of embedding classical computation into the quantum circuit model. This leads to simple extensions of Clifford circuits which are classically simulatable. These circuits can be efficiently simulated by classical sampling ('weak simulation') even though the problem of exactly computing the outcomes of measurements for these circuits ('strong simulation') is proved to be #P-complete--thus showing that there is a separation between weak and strong classical simulation of quantum computation.

arXiv:0811.0898Nov 6, 200831 citations

Classical simulation of quantum computation, the gottesman-Knill theorem, and slightly beyond

M. Van den Nest

We study classical simulation of quantum computation, taking the Gottesman-Knill theorem as a starting point. We show how each Clifford circuit can be reduced to an equivalent, manifestly simulatable circuit (normal form). This provides a simple proof of the Gottesman-Knill theorem without resorting to stabilizer techniques. The normal form highlights why Clifford circuits have such limited computational power in spite of their high entangling power. At the same time, the normal form shows how the classical simulation of Clifford circuits fits into the standard way of embedding classical computation into the quantum circuit model. This leads to simple extensions of Clifford circuits which are classically simulatable. These circuits can be efficiently simulated by classical sampling (``weak simulation'') even though the problem of exactly computing the outcomes of measurements for these circuits (``strong simulation'') is proved to be $\#\mathbf{P}$ -complete---thus showing that there is a separation between weak and strong classical simulation of quantum computation.We study classical simulation of quantum computation, taking the Gottesman-Knill theorem as a starting point. We show how each Clifford circuit can be reduced to an equivalent, manifestly simulatable circuit (normal form). This provides a simple proof of the Gottesman-Knill theorem without resorting to stabilizer techniques. The normal form highlights why Clifford circuits have such limited computational power in spite of their high entangling power. At the same time, the normal form shows how the classical simulation of Clifford circuits fits into the standard way of embedding classical computation into the quantum circuit model. This leads to simple extensions of Clifford circuits which are classically simulatable. These circuits can be efficiently simulated by classical sampling (``weak simulation'') even though the problem of exactly computing the outcomes of measurements for these circuits (``strong simulation'') is proved to be $\#\mathbf{P}$-complete---thus showing that there is a separation between weak and strong classical simulation of quantum computation.

Quantum Information and ComputationMar 1, 201082 citations

Generalised Clifford groups and simulation of associated quantum circuits

Sean Clark, Richard Jozsa, Noah Linden

Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G tensor G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2 we exhaustively treat all finite subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups G_n generated by X and the n^th root of Z.Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G tensor G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2 we exhaustively treat all finite subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups G_n generated by X and the n^th root of Z.

arXiv:quant-ph/0701103Jan 15, 20074 citations

Generalized Clifford groups and simulation of associated quantum circuits

Stephen R. L. Clark, Richard Jozsa, Noah Linden

Quantum computations starting with computational basis states and involving only Clifford operations, are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup $G$ of $U(d)$ . In particular we seek $G$ such that $G\otimes G$ has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case $d = 2$ we exhaustively treat all finite irreducible subgroups of $U(2)$ and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the $n^{\rm th}$ root of $Z$ for $n \in \N$ .Quantum computations starting with computational basis states and involving only Clifford operations, are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup $G$ of $U(d)$. In particular we seek $G$ such that $G\otimes G$ has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case $d = 2$ we exhaustively treat all finite irreducible subgroups of $U(2)$ and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the $n^{\rm th}$ root of $Z$ for $n \in \N$.

Quantum Information and ComputationJan 1, 200822 citations

The Heisenberg representation of quantum operators provides a powerful technique for reasoning about quantum circuits, albeit those restricted to the common (non-universal) Clifford set

H

S

and

CNOT

\#\mathbf{P}

-complete---thus showing that there is a separation between weak and strong classical simulation of quantum computation.

G

U(d)

. In particular we seek

G

such that

G\otimes G

has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case

d = 2

we exhaustively treat all finite irreducible subgroups of

U(2)

and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the

n^{\rm th}

root of

Z

for

n \in \N