HZH-Routine

Looking at each part of a program individually is the usual approach when we learn a new programming language. So, if you look closely at a Quantum Circuit,A quantum circuit is a sequence of quantum gates applied to qubits, representing the operations in a quantum computation. Each gate changes the qubits’ state using quantum mechanics principles like superposition and entanglement. The final qubit states, when measured, yield the circuit’s computational result probabilistically.
Learn more about Quantum Circuit you will see a chain of Quantum GatesA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
Learn more about Quantum Gate connected in series. Most examples can even be reduced to three simple Quantum GatesA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
Learn more about Quantum Gate that reflect the Pauli OperatorsA Pauli operator is one of three 2×2 complex matrices — **σₓ, σᵧ, σ_z** — that represent the basic quantum spin operations on a single qubit. They correspond to rotations or measurements along the x, y, and z axes of the Bloch sphere. Together with the identity matrix, they form a basis for all single-qubit operations in quantum mechanics.
Learn more about Pauli Operator plus the Controlled-NOT Operator.The CNOT (Controlled-NOT) operator is a two-qubit quantum gate that flips the **target qubit** (applies an X gate) only if the **control qubit** is in the state (|1⟩). In matrix form, it leaves (|00⟩) and (|01⟩) unchanged, but swaps (|10⟩) and (|11⟩). It’s essential for creating entanglement between qubits.
Learn more about Controlled-NOT Operator Each Quantum GateA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
Learn more about Quantum Gate seems to have a fixed purpose: X-GateThe X-Gate flips a qubit's probability amplitudes between its two basis states, effectively exchanging their roles. It doesn’t just turn into , but rotates any quantum state halfway around the x-axis of the Bloch sphere. This means it inverts the qubit's orientation while preserving the overall shape of its state on the sphere.
Learn more about X-Gate flips a Bit,A bit (short for “binary digit”) is the smallest unit of data in computing, representing a value of either 0 or 1. It’s the fundamental building block of all digital information. Multiple bits combine to form larger units like bytes (8 bits) and encode more complex data such as numbers, text, or images.
Learn more about Binary Digit Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate flips a Quantum Phase,A **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase Y GateY Gate
Learn more about Y Gate does both, and Controlled-NOT OperatorThe CNOT (Controlled-NOT) operator is a two-qubit quantum gate that flips the **target qubit** (applies an X gate) only if the **control qubit** is in the state (|1⟩). In matrix form, it leaves (|00⟩) and (|01⟩) unchanged, but swaps (|10⟩) and (|11⟩). It’s essential for creating entanglement between qubits.
Learn more about Controlled-NOT Operator entangles Qubits.A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit This seems clear and predictable. Until it doesn't.

This post is accompanied by a PDF file summarizing the key points.

Unfortunately, this approach does not work well with Quantum Circuits.A quantum circuit is a sequence of quantum gates applied to qubits, representing the operations in a quantum computation. Each gate changes the qubits’ state using quantum mechanics principles like superposition and entanglement. The final qubit states, when measured, yield the circuit’s computational result probabilistically.
Learn more about Quantum Circuit At first sight, the Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate often seems to have no effect. But later your MeasurementIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement results change. You observe effects that do not match the simple explanation. It feels as if the Quantum ComputerA quantum computer is typically a large, highly controlled system kept at near-absolute-zero temperatures to preserve quantum behavior. It contains a processor with qubits—often made from superconducting circuits, trapped ions, or photons—manipulated by microwaves, lasers, or magnetic fields. Surrounding systems handle cooling, error correction, and control electronics to maintain quantum coherence and read out results.
Learn more about Quantum Computer is behaving incorrectly and the Quantum GatesA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
Learn more about Quantum Gate are not doing what they should.

But the problem isn’t with the machine. It’s with how we think about Quantum GatesA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
Learn more about Quantum Gate. What if a Quantum Gate'sA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
Learn more about Quantum Gate meaning depends not on what it is, but on the basis you’re looking from?

This small shift in perspective is one of the biggest conceptual jumps in Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing Preskill, J., 2018, CreateSpace Independent Publishing Platform, . And the simplest proof of it is hidden inside a three-letter sequence:

But before we name what this really means, let’s first unpack why this is true. And why it matters far beyond one equation.

Cracks in the Old Belief

Start with the invisible Z-Gate.The Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate

At first glance, that looks boring. No flipping, no mixing, just a PhaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase change. If you measuredIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit right after, you’d never know anything happened. Because the minus sign doesn't matter for the Measurement.In quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement But that minus sign matters for Interference.Interference in quantum computing refers to the way probability amplitudes of quantum states combine—sometimes reinforcing each other (constructive interference) or canceling out (destructive interference). Quantum algorithms exploit this to amplify the probability of correct answers while suppressing incorrect ones. It’s a key mechanism that gives quantum computers their computational advantage.
Learn more about Interference It determines how AmplitudesIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
Learn more about Amplitude add or cancel.

So the Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate isn’t doing nothing. It’s rotating your Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State around the-axis of the Bloch SphereThe Bloch sphere is a geometric representation of a single qubit’s quantum state as a point on or inside a unit sphere. The north and south poles represent the classical states |0⟩ and |1⟩, while any other point corresponds to a superposition of them. Its position encodes the qubit’s relative phase and probability amplitudes, making it a visual tool for understanding quantum state evolution.
Learn more about Bloch Sphere. Something you can’t see from the standard MeasurementIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement view but that has very real effects.

On its own, the Hadamard GateThe Hadamard operator, often denoted H, is a single-qubit quantum gate that creates an equal superposition of and . In other words, It turns the states of the computational basis and into the states of the Fourier basis and .
Learn more about Hadamard Operator looks innocent too.

Think of the Hadamard OperatorThe Hadamard operator, often denoted H, is a single-qubit quantum gate that creates an equal superposition of and . In other words, It turns the states of the computational basis and into the states of the Fourier basis and .
Learn more about Hadamard Operator like rotating a cube. The cube hasn’t changed, but you now see a different face. The Hadamard OperatorThe Hadamard operator, often denoted H, is a single-qubit quantum gate that creates an equal superposition of and . In other words, It turns the states of the computational basis and into the states of the Fourier basis and .
Learn more about Hadamard Operator does that for QubitsA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit. It changes which axes you’re using to describe the same state.

In ?, we create a custom Quantum GateA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
Learn more about Quantum Gate that applies the -sequence on a Qubit.A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit

How To Create Custom Operator With Qiskit

Stop Writing Quantum Spaghetti Code

October 29, 2025•5 min

Simply stacking quantum gates on one another is the fastest way to turn your code into a confusing mess

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from qiskit import QuantumCircuit

 

# Create a circuit for the hzh gate

hzh_qc = QuantumCircuit(1, name="HZH")

 

# change perspective (into Fourier basis)

hzh_qc.h(0)

 

# flip the phase of |1>

hzh_qc.z(0)

 

# change perspective (back into computational basis)

hzh_qc.h(0)

 

# turn the circuit into a gate

HzhGate = hzh_qc.to_gate(label="HZHGate")

This code listing
1. imports the required QuantumCircuit class from Qiskit,
2. creates a QuantumCircuit with a single QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
   Learn more about Quantum Bit
3. turns the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
   Learn more about Quantum Bit from the Computational BasisThe computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
   Learn more about Computational Basis into the Fourier BasisThe Fourier basis is a set of sine and cosine functions that can represent any periodic signal as a weighted sum of these functions. Each basis function corresponds to a specific frequency, capturing how much of that frequency is present in the signal. In essence, it’s the coordinate system for expressing signals in terms of their frequency components instead of time.
   Learn more about Fourier Basis
4. applies the Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
   Learn more about Z-Gate that flips the phase of the is a basis state.
   Learn more about -AmplitudeIn quantum computing an amplitude is a complex number that describes the weight of a basis state in a quantum superposition. The squared magnitude of an amplitude gives the probability of measuring that basis state. Amplitudes can interfere, this means adding or canceling, allowing quantum algorithms to bias outcomes toward correct solutions.
   Learn more about Amplitude, effectively turning the qubit from to
5. turns the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
   Learn more about Quantum Bit back from the Fourier BasisThe Fourier basis is a set of sine and cosine functions that can represent any periodic signal as a weighted sum of these functions. Each basis function corresponds to a specific frequency, capturing how much of that frequency is present in the signal. In essence, it’s the coordinate system for expressing signals in terms of their frequency components instead of time.
   Learn more about Fourier Basis into the Computational BasisThe computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
   Learn more about Computational Basis
6. turns the QuantumCircuit into a custom Quantum GateA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
   Learn more about Quantum Gate

So, we can say:

Quantum GatesA quantum gate is a basic operation that changes the state of one or more qubits, similar to how a logic gate operates on bits in classical computing. It uses unitary transformations, meaning it preserves the total probability (the state’s length in complex space). Quantum gates enable superposition and entanglement, allowing quantum computers to perform computations that classical ones cannot efficiently replicate.
Learn more about Quantum Gate don’t have fixed meanings. Their effect depends on the basis you’re looking in. That’s what the routine demonstrates so cleanly. A Z-GateThe Z-gate changes the phase of the state by (or radians) while leaving the state unchanged. This means the amplitude of gains a negative sign, effectively flipping its phase. It's a phase-flip operation that rotates the quantum state vector a half turn around the Z-axis of the Bloch sphere.
Learn more about Z-Gate is a PhaseA **quantum phase** is the angle component of a particle’s wavefunction that determines how its probability amplitude interferes with others. It doesn’t affect observable probabilities directly but becomes crucial when comparing two or more states, as phase differences lead to interference effects. Essentially, it encodes the relative timing or “alignment” of quantum waves.
Learn more about Quantum Phase flip around one axis; surround it with Hadamard Operators,The Hadamard operator, often denoted H, is a single-qubit quantum gate that creates an equal superposition of and . In other words, It turns the states of the computational basis and into the states of the Fourier basis and .
Learn more about Hadamard Operator and you’re now looking around a different axis. Then, it becomes a BitA bit (short for “binary digit”) is the smallest unit of data in computing, representing a value of either 0 or 1. It’s the fundamental building block of all digital information. Multiple bits combine to form larger units like bytes (8 bits) and encode more complex data such as numbers, text, or images.
Learn more about Binary Digit flipNielsen, M.A., 2010, Cambridge university press, .

Sounds abstract? It is. Until you picture it geometrically as shown in ?. On the Bloch Sphere,The Bloch sphere is a geometric representation of a single qubit’s quantum state as a point on or inside a unit sphere. The north and south poles represent the classical states |0⟩ and |1⟩, while any other point corresponds to a superposition of them. Its position encodes the qubit’s relative phase and probability amplitudes, making it a visual tool for understanding quantum state evolution.
Learn more about Bloch Sphere the HZH sequence moves the Quantum State VectorA quantum state vector is a mathematical object (usually denoted |ψ⟩) that fully describes the state of a quantum system. Its components give the probability amplitudes for finding the system in each possible basis state. The squared magnitude of each component gives the probability of measuring that corresponding outcome.
Learn more about Quantum State Vector in a well-defined way starting from is a basis state.
Learn more about and ending in is a basis state.
Learn more about .